ROBINSON'S  MATHEMATICAL  SERIES. 


CONIC  SECTIONS 


AND 


ANALYTICAL  GEOMETRY; 


THEORETICALLY  AND  PRACTICALLY  ILLUSTRATED, 


BY 

HORATIO  N.  ROBINSON,  LL.D., 

LATE  PROFESSOR  OF  MATHEMATICS  IN  THE  U.  8.  NAVY,   AND  AUTHOR  OF  A  FULL 
OF  MATHEMATICS. 


NEW    YOEK: 
IVISON,     PHINNEY    &     COMPANY, 

48    &     50     WALKER     STREET. 

CHICAGO: 

S.    C.     GRIGGS    &    COMPANY, 

89   &   41    LAKE    STBEET. 

1863. 


R 


Engineering 
Library 

OBINSON'S 


u?LET$,  M,ost  ERA.CTICAL,  and  most  SCIENTIFIC  SERIES  of 
MATHEMATICAL  TEXT-BOOKS  ever  issued  in  this  country. 


I.    Bobinson's  Progressive  Table  Book, $    12 

II.    Bobinson's  Progressive  Primary  Arithmetic,  -       -       -       -       15 

III.  Bobinson's  Progressive  Intellectual  Arithmetic,  25 

IV.  Bobinson's  Budiments  of  Written  Arithmetic,  25 
V.    Bobinson's  Progressive  Practical  Arithmetic,  56 

VI.    Bobinson's  Key  to  Practical  Arithmetic, 50 

VII.    Bobinson's  Progressive  Higher  Arithmetic,  75 

VIII.    Bobinson's  Key  to  Higher  Arithmetic,  »  75 

IX.    Bobinson's  New  Elementary  Algebra,      -----       75 

X.    Bobinson's  Key  to  Elementary  Algebra, 75 

XI.    Bobinson's  University  Algebra, 1  25 

XII.    Bobinson's  Key  to  University  Algebra, 1  00 

XIII.  Bobinson's  New   University  Algebra,        -       -       -       -       -    1  50 

XIV.  Bobinson's  Key  to  New  University  Algebra,  •        -       -       -    1  25 
XV.    Bobinson's  New  Geometry  and  Trigonometry,     -       -       -    1  50 

XVI.    Bobinson's  Surveying  and  Navigation,     -       -       -       -       -    1  50 

XVII.    Bobinson's  Analyt.  Geometry  and  Conic  Sections,      -       -    1  50 

XVIII.    Bobinson's  DifFeren.  and  Int.  Calculus,  (in  preparation,)-       -    1  50 

XIX.    Bobinson's  Elementary  Astronomy, 75 

XX.    Bobinson's  University  Astronomy,     -       -       -       -       -       -175 

XXI.    Bobinson's  Mathematical  Operations, 2  25 

XXII.    Bobinson's  Key  to   Geometry  and  Trigonometry,  Conic 

Sections  and  Analytical  Geometry, l  50 


Entered,  according  to  Act  of  Congress,  in  the  year  1860,  by 
HOKATIO    N.     EOBINSON,    LL.D., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Northern 
District  of  New  York. 


PREFACE. 


In  the  preparation  of  the  following  work  the  object  has  been  to 
bring  within  the  compass  of  one  volume  of  convenient  size  an  ele 
mentary  treatise  on  both  Conic  Sections  and  Analytical  Geometry. 

In  the  first  part,  the  properties  of  the  curves  known  as  the  Conic 
Sections  are  demonstrated,  principally  by  geometrical  methods  ;  that 
is,  in  the  investigations,  the  curves  and  parts  connected  with  them 
are  constantly  kept  before  the  mind  by  their  graphic  representations, 
and  we  reason  directly  upon  them. 

In  the  purely  Analytical  Geometry  the  process  is  quite  different. 
Here  the  geometrical  magnitudes,  themselves,  or  those  having  cer 
tain  relations  to  them,  are  represented  by  algebraic  symbols,  and  we 
seek  to  express  properties  and  imposed  conditions  by  means  of  these 
symbols.  The  mind  is  thus  relieved,  in  a  great  measure,  of  the  ne 
cessity  of  holding  in  view  the  often-times  complex  figures  required 
in  the  intermediate  steps  of  the  first  method.  It  is,  mainly,  at  the 
beginning  and  end  of  our  investigations  that  we  have  to  deal  with 
concrete  quantity.  That  is,  after  we  have  expressed  known  and  im 
posed  conditions,  analytically,  our  reasoning  is  independent  of  the 
kind  of  quantity  involved,  until  the  conclusion  is  reached  in  the 
form  of  an  algebraic  expression,  which  must  then  receive  its  geo 
metrical  interpretation. 

Much  of  the  value  of  Analytical  Geometry,  as  a  disciplinary 
study,  will  be  derived  from  a  careful  consideration,  in  each  case,  of 
this  process  of  passing  from  the  concrete  to  the  abstract  and  the 

7940O7  (ff) 


iv  PREFACE. 

converse,  and  both  teacher  and  student  are  earnestly  recommended 
to  give  it  a  large  share  of  their  attention. 

In  both  divisions  of  the  work  the  object  has  been  to  present  the 
subjects  in  the  simplest  manner  possible,  and  hence,  in  the  first, 
analytical  methods  have  been  employed  in  several  propositions  when 
results  could  be  thereby  much  more  easily  obtained;  and  for  the 
same  reason,  in  the  second  division,  a  few  of  the  demonstrations  are 
almost  entirely  geometrical. 

The  analytical  part  terminates,  with  the  exception  of  some  exam 
ples,  with  the  Chapter  on  Planes.  Three  others  might  have  been 
added ;  one  on  the  transformation  of  Co-ordinates  in  Space,  another 
on  Curves  in  Space,  and  a  third  on  Surfaces  of  Revolution  and 
curved  surfaces  in  general :  but  the  work,  as  it  is,  covers  more 
ground  than  is  generally  gone  over  in  Schools  and  Colleges,  and  is 
sufficiently  extensive  for  the  wants  of  elementary  education.  Nu 
merous  examples  are  given  under  the  several  divisions  in  the  second 
part  to  illustrate  and  impress  the  principles. 

The  Author  has  great  pleasure  in  acknowledging  his  obligations 
to  Prof.  I.  F.  Quinby,  A.  M.,  of  the  University  of  Rochester,  N.  Y., 
formerly  Assistant  Prof,  of  Mathematics  in  the  United  States  Mili 
tary  Academy,  at  West  Point,  for  valuable  services  rendered  in  the 
preparation  of  this  treatise,  as  well  as  for  the  contribution  to  it  of 
much  that  is  valuable  both  in  matter  and  arrangement.  His  thor 
ough  scholarship,  as  well  as  his  long  and  successful  experience  as  an 
instructor  in  the  class-room,  preeminently  qualified  him  to  perform 
such  labor. 

December,  1861. 


CONTENTS. 

CONIC    SECTIONS. 

DEFINITIONS. 

Conical  Surfaces, PAGE  9 

Conic  Sections, 10 

THE   ELLIPSE. 

Definitions  and  Explanations, 11 

Propositions  relating  to  the  Ellipse, 13 

THE    PARABOLA. 

Definitions  and  Explanations, 41 

Propositions  relating  to  the  Parabola, 43 

THE   HYPERBOLA. 

Definitions  and  Explanations, 65 

Propositions  relating  to  the  Hyperbola, 67 

ASYMPTOTES. 

Definition, 91 

Propositions  establishing  relations  between  the  Hyperbola  and 

its  Asymptotes, 91 

1* 


VI 


CONTENTS. 


ANALYTICAL    GEOMETRY. 

General  Definitions  and  Remarks, 


GENERAL  PROPERTIES  OF  GEOMETRICAL  MAGNITUDES, 
CHAP  TEE   I. 

OF  POSITIONS  AND  STRAIGHT  LINES  IN  A  PLANE 
AND  THE  TRANSFORMATION  OF  CO-ORDINATES. 

Definitions  and  Explanations, ,    97 

Propositions  relating  to  Straight  Lines  in  a  Plane, 100 

Transformation  of  Co-ordinates, 119 

Polar  Co-ordinates, 122 

CHAPTER   II. 

THE    CIRCLE. 

LINES  OF  THE  SECOND  ORDEB. 

Propositions  relating  to  the  Circle 124 

Polar  equation  of  the  Circle, 132 

Application  in  the  solution  of  Equations  of  the  second  degree,  134 
Examples, 139 

CHAPTER   III. 

THE    ELLIPSE. 

The  description  of  the  Ellipse  and  Propositions  establishing 

its  properties, 140 

Example, 167 

CHAPTER   IV. 
THE    PARABOLA. 

The  description  of  the  Parabola  and  propositions  establishing 

its  properties, 169 

Polar  equation  of  the  Parabola, 183 

Application  in  the  solution  of  equations  of  the  second  degree,  185 
Examples 187 


CONTENTS.  vii 

CHAPTER  Y. 
THE    HYPERBOLA. 

The  Description  of  the  Curve,  and  Propositions  Establishing 

its  Properties, 188 

ASYMPTOTES  OF  THE  HYPERBOLA. 

Definition  and  Explanation, 201 

The  Equation  of  the  Hyperbola  referred  to  its  Asymptotes,  and 

Properties  deduced  therefrom, 202 

CHAPTER    VI. 

ON  THE  GEOMETRICAL  REPRESENTATION  OF 
EQUATIONS  OF  THE  SECOND  DEGREE  BE 
TWEEN  TWO  VARIABLES. 

Object  of  the  Discussion, 210 

Solution  and  Discussion  of  the  General  Equation, 211 

Criteria  for  the  Interpretation  of  any  Equation  of  the  Second 

Degree  between  two  Variables, 221 

APPLICATIONS. 

First,  B*—±AC<Q,  the  Ellipse, 222 

Second,  B*—±A  <7>0,  the  Hyperbola, 226 

Third,  £*— 4^O=0,  the  Parabola, 231 

Examples, 233 

CHAPTER  VII. 

On  the  Intersection  of  Lines,  and  the  Geometrical  Solution  of 

Equations, 237 

Remarks  on  the  Interpretation  of  Equations, 244 


viil  CONTENTS. 

CHAPTEE  YIII. 
STRAIGHT    LINES    IN    SPACE. 

Co-ordinate  Planes  and  Axes, 249 

The  Equations  and  Relations  of  Straight  Lines  in  Space, ....   250 

CHAPTEE    IX. 
ON  THE  EQUATION  OF  A  PLANE. 

The  Equations  and  Relations  of  Planes, 258 

Examples  Relating  to  Straight  Lines  in  Space  and  to  Planes, .   269 
Miscellaneous  Examples, 273 


CONIC   SECTIONS. 


DEFINITIONS. 

1.  A  Conical  Surface,  or  a  Cone  is,  in  its  general  accept 
ation,  the  surface  that  is  generated  by  the  motion  of  a 
straight  line  of  indefinite  extent,  which  in  its  different 
positions    constantly  passes  through  a  fixed  point  and 
touches  a  given  curve. 

The  moving  line  is  called  the  generatrix,  the  curve  that 
it  touches  the  directrix,  the  fixed  point  the  vertex,  and  the 
generatrix  in  any  of  its  positions  an  element,  of  the  cone. 

The  generatrix  in  all  its  positions  extending  without 
limit  beyond  the  vertex  on  either  side,  will  by  its  motion 
generate  two  similar  surfaces  separated  by  the  vertex, 
called  the  nappes  of  the  cone. 

2.  The  Axis  of  a  cone  is  the  indefinite  line  passing 
through  the  vertex  and  the  center  of  the  directrix. 

3.  The  intersection  of  the  cone  by  any  plane  not  pass 
ing  through  its  vertex,  that  cuts  all  its  elements,  may  be 
taken  as  the  directrix ;  and  when  we  regard  the  cone  as 
limited  by  such  intersection,  it  is  called  the  base  of  the  cone. 
If  the  axis  is  perpendicular  to  the  plane  of  the  base,  the 
cone  is  said  to  be  right;  and  if  in  addition  the  base  is  a 
circle,  we  have  a  right  cone  with  a  circular  base.    This  is  the 
same  as  the  cone  defined  in  Geometry,  (Book  VII,  Def. 
16),  and  in  the  following  pages  it  is  to  be  understood  that 

all  references  are  made  to  it,  unless  otherwise  stated. 

(9) 


10-  GONIC    SECTIONS. 

4.  Conic  Sections  are  the  figures  made  by  a  plane  cutting 
a  cone. 

5.  There  are  five  different  figures  that  can  be  made  by 
a  plane  cutting   a  cone,  namely:    a  triangle,  a  circle,  an 
ellipse,  a  parabola,  and  an  hyperbola. 

HE  MARK.  The  three  last  mentioned  are  commonly  regarded  as 
embracing  the  whole  of  conic  sections ;  but  with  equal  propriety  the 
triangle  and  the  circle  might  be  admitted  into  the  same  family.  On 
the  other  hand  we  may  examine  the  properties  of  the  ellipse,  the 
parabola,  and  the  hyperbola,  in  like  manner  as  we  do  a  triangle  or  a 
circle,  without  any  reference  whatever  to  a  cone. 

It  is  important  to  study  these  curves,  on  account  of  their  exten 
sive  application  to  astronomy  and  other  sciences. 

6.  If  a  plane  cut  a  cone  through  its  vertex,  and  termin 
ate  in  any  part  of  its  base,  the  section  will  evidently  be  a 
triangle. 

7.  If  a  plane  cut  a  cone  parallel  to  its  base,  the  section 
will  be  a  circle. 

8.  If  a  plane  cut  a  cone  obliquely  through  all  of  the 
elements,  the  section  will  represent  a  curve  called   an 
ellipse. 

9.  If  a  plane  cut  a  cone  parallel  to  one  of  its  elements, 
or  what  is  the  same  thing,  if  the  cutting  plane  and  an 
element  of  the  cone  make  equal  angles  with  the  base,  then 
the  section  will  represent  a  parabola. 

10.  If  a  plane  cut  a  cone,  making  a  greater 
angle  with  the  base  than  the  element  of  the 
cone  makes,  then  the  section  is  an  hyperbola. 

11.  And  if  the  plane  be  continued  to  cut 
the  other  nappe  of  the  cone,  this  latter  inter 
section  will  be  the  opposite  hyperbola  to  the 
former. 

12.  The  Vertices  of  any  section  are  the  points  where  the 
cutting  plane  meets  the  opposite  elements  of  the  cone, 
or  the  sides  of  the  vertical  triangular  section,  as  A  and  B. 


THE    ELLIPSE.  11 

Hence,  the  ellipse  and  the  opposite  hyperbo 
las  have  each  two  vertices;  but  the  parabola 
has  only  one,  unless  we  consider  the  other  as 
at  an  infinite  distance. 

13.  The  Axis,  or  Transverse  Diameter  of  a  conic 
section,  is  the  line  or  distance  AB  between  the 
vertices. 

Hence,  the  axis  of  a  parabola  is  infinite  in  length,  AB 
being  only  a  part  of  it. 

The  properties  of  the  three  curves  known  as  the  Conic 
Sections  will  first  be  investigated  without  any  reference 
to  the  cone  whatever ;  and  afterward  it  will  be  shown  that 
these  curves  are  the  several  intersections  of  a  cone  by  a 
plane. 

THE  ELLIPSE. 
DEFINITIONS. 

1.  The  Ellipse  is  a  plane  curve  described  by  the  motion 
of  a  point  subjected  to  the  condition  that  the  sum  of  its  dis 
tances  from  two  fixed  points  shall  be  constantly  the  same. 

2.  The  two  fixed  points  are  called 
the  foci.      Thus  F,  F',  we  foci. 

3.  The  Center  is  the  point  (7,  the 
middle  point  between  the  foci. 

4.  A  Diameter  is  a  straight  line 

through  the  center,  and  terminated  both  ways  by  the 
curve. 

5.  The  extremities  of  a  diameter  are  called  its  vertices. 
Thus,  DD*  is  a  diameter,  and  JD  and  Df  are  its  vertices. 

6.  The  Major,  or  Transverse  Axis,  is  the  diameter  which 
passes  through  the  foci.     Thus,  AAe  is  the  major  axis. 

7.  The  Minor,  or  Conjugate  Axis  is  the  diameter  at  right 


12  CONIC    SECTIONS. 

angles  to  the  major  axis.     Thus,  CE  is  the  semi  minor 
axis. 

8.  The  distance  between  the  center  and  either  focus  is 
called  the  eccentricity  when  the  semi  major  axis  is  unity. 

That  is,  the  eccentricity  is  the  ratio  between  CA  and 

CF 

CF;  or  it  is  -^7  ;  hence,  it  is  always  less  than  unity. 

The  less  the  eccentricity,  the  nearer  the  ellipse  approaches 
the  circle. 

9.  A  Tangent  is  a  straight  line  which  meets  the  curve 
in  one  point  only;  and,  being  produced,  does  not  cut  it. 

10.  A  Normal  to  a  curve  at  any  point  is  a  perpendicular 
to  the  tangent  at  that  point. 

11.  An  Ordinate  to  a  Diameter  is  a  straight  line  drawn 
from  any  point  of  the  curve  to  the  diameter,  parallel  to  a 
tangent  passing  through  one  of  the  vertices  of  that  diame 
ter. 

REMARK. — A  diameter  and  its  ordinate  are  not  at  right  angles, 
unless  the  diameter  be  either  the  major  or  minor  axis. 

12.  The  parts  into  which  a  diameter  is  divided  by  an 
ordinate,  are  called  abscissas. 

13.  Two  diameters  are  said  to  be  conjugate,  when  either 
is  parallel  to  the  tangent  lines  at  the  vertices  of  the  other. 

14.  The  Parameter  of  a  diameter  is  a  third  proportional 
to  that  diameter  and  its  conjugate. 

15.  The  paramater  of  the  major  axis  is  called  fhe  prin- 
cipal  parameter,  or  latus  rectum  ;  and,  as  will  be  proved,  is 
equal  to  the  double  ordinate  through  the  focus.     Thus 
Ff  G-  is  one  half  of  the  principal  parameter. 

16.  A  Sub-tangent  is  that  part  of  the  axis  produced,  which 
is  included  between  a  tangent  and  the  ordinate,  drawn 
from  the  point  of  contact. 

17.  A  Sub-normal  is  that  part  of  the  axis  which  is  includ 
ed  between  the  normal  and  the  ordinate,  drawn  from  the 
point  of  contact. 


THE    ELLIPSE. 


13 


PROPOSITION  I.    PROBLEM. 

To  describe  an  Ellipse. 

Assume  any  two  points,  as  F  and 
F'  and  take  a  thread  longer  than 
the  distance  between  these  points,  A 
fastening  one  of  its  extremities  at 
the  point  F  and  the  other  at  the 
point  F' .  Now  if  the  point  of  a  pencil  be  placed  in  the 
loop  and  moved  entirely  around  the  points  F  and  F* ',  the 
thread  being  constantly  kept  tense,  it  will  describe  a  curve 
as  represented  in  the  adjoining  figure,  and,  by  definition 
1,  this  curve  is  an  ellipse. 


PROPOSITION   II.-THEOREM. 

The  major  axis  of  an  ellipse  is  equal  to  the  sum  of  the  two 
lines  drawn  from  any  point  in  the  curve  to  the  foci. 

Suppose  the  point  of  a  pencil  at 
D  to  move  along  in  the  loop,  hold 
ing  the  threads  F'D  and  FD  at  A'( 
equal  tension  ;  when  D  arrives  at 
A,  there  will  be  two  lines  of  threads 
between  F  and  A.  Hence,  the  entire  length  of  the  threads 
will  be  measured  by  F'F+%FA.  Also,  when  D  arrives 
at  A',  the  length  of  the  threads  is  measured  by  FF'+ 
ZF'A'. 

Therefore,      .     FFf+ZFA=FF'+2F'  A'  i 

Hence,     ...       .   FA=F'  A' 

From  the  expression  FF'+^FA,  take  away  FA,  and 
add  F'  A',  and  the  sum  will  not  be  changed,  and  we  have 


Therefore,       .      F'D+FD=A'A 
Hence  the  theorem  ;  the  major  axis  of  an  ellipse,  etc. 
2 


14 


CONIC    SECTIONS. 


PEOPOSITION  III .— T  H  E  0  B  E  M 

An  ellipse  is  bisected  by  either  of  its  axes. 

Let  F,  F*  be  the  foci,  AA'  the  ma 
jor  and  BBf  the  minor  axis  of  an 
ellipse ;  then  will  either  of  these  A 
axis  divide  the  ellipse  into  equal 
parts. 

Take  any  point,  as  P  in  the  el 
lipse,  and  from  this  point  draw  ordinates,  one  to  the  ma 
jor  and  another  to  the  minor  axis,  and  produce  these  or 
dinates,  the  first  to  P',  the  second  to  P",  making  the  parts 
produced  equal  to  the  ordinates  themselves.  It  is  evident 
that  the  proposition  will  be  established  when  we  have 
proved  that  P'  and  P'  are  points  of  the  curve. 

First.  Fis  a  point  in  the  perpendicular  to  PPf  at  its 
middle  point;  therefore  FP'=FP  (Scho.  1,  Th.  18,  B.  1 
Geom.)  for  the  same  reason  F'Pf=FfP. 

Whence,  by  addition, 

FP'+FfP'=FP+FfP. 

That  is,  the  sum  of  the  distances  from  P'  to  the  foci  is 
equal  to  the  sum  of  the  distances  from  P  to  the  foci ;  but 
by  hypothesis  Pis  a  point  of  the  ellipse;  therefore  Pr  is 
also  a  point  of  the  ellipse,  (Def.  1). 

Second.  The  trapezoids  P"dCF',  PdCF  are  equal,  be 
cause  F'C=FC,  dP'=dP  by  construction,  and  the  angles 
at  d  and  C  in  each  are  equal,  being  right  angles ;  these 
figures  will  therefore  coincide  when  applied,  and  we  have 
P'F'  equal  to  PF  and  the  angle  P'F'F  equal  to  the  angle 
PFF'.  Hence  the  triangles  P'F'F,  PFFf  are  equal  hav 
ing  the  two  sides  P'F',  F'Fsmd  the  included  angle  P"F'F 
in  the  one  equal,  each  to  each  to  the  two  sides  PF,  FFf 
and  the  included  angle  PFF'  in  the  other. 

Therefore,  P'F'+P"F=PF'+F  P 

That  is,  the  sum  of  the  distances  from  P"  to  the  foci  is 


THE    ELLIPSE.  15 

equal  to  the  sum  of  the  distances  from  P  to  the  foci,  and 
since  P  is  a  point  of  the  ellipse  P"  must  also  be  found  on 
the  ellipse. 

Hence  the  theorem ;  an  ellipse  is  bisected,  etc. 


PROPOSITION  IV.—  THEOREM. 

The  distance  from  either  focus  of  an  ellipse  to  the  extremity 
of  the  minor  axis  is  equal  to  the  semi-major  axis. 

Let  A  A1  be  the  major  axis,  J^and 
F'  the  foci,  and  CD  the  semi-minor 
axis  of  an  ellipse  ;  then  will  FD=  A'| 
F'D  be  equal  to  CA. 

Because  F'C=  CF  and  CD  is  at 
right  angles  to  F'F,  we  have  FfD=FD. 

But,  F'D+FD=A'A 

Or,  2FD=A'A 

Therefore,  FD=±A'A,  or  CA. 

Hence  the  theorem  ;  the  distance  from  either  focus,  etc. 

SCHOLIUM.  —  The  half  of  the  minor  axis  is  a  mean  proportional 
between  the  distance  from  either  focus  to  the  principal  vertices. 

In  the  right-angled  triangle  FCD  we  have 


But,  FD=AC 

Therefore,  ~CD2=AG^—  FO* 

=  (AC+FC}(A  C—FC) 

=AF'XAF 
Or,  AF:  CD=CD  :  FA' 

PROPOSITION   V.—  THEOREM 

Every  diameter  of  an  ellipse  is  bisected  at  the  center. 

Let  D  be  any  point  in  the  curve,  and  C  the  center. 
Draw  DC,  and  produce  it.     From  F'  draw  jF7'  17  parallel; 


16 


CONIC    SECTIONS. 


to  FD;  and  from  F  draw  FD'  par 
allel  to  F'D.  The  figure  DFD'F'  is 
a  parallelogram  by  construction;  and 
therefore  its  opposite  sides  are  equal. 

Hence,  the  sum  of  the  two  sides 
F'D'  and  D'Fis  equal  to  F'D  and  DF;  therefore,  by  def 
inition  1,  the  point  Df  is  in  the  ellipse.  But  the  two  di 
agonals  of  a  parallelogram  bisect  each  other ;  therefore, 
DC=CDr,  and  the  diameter  DDr  is  bisected  at  the  center, 
(7,  and  DD'  represents  any  diameter  whatever. 

Hence  the  theorem ;  every  diameter,  etc. 

Cor.  The  quadrilateral  formed  by  drawing  lines  from 
the  extremities  of  a  diameter  to  the  foci  of  an  ellipse,  is 
a  parallelogram. 


PROPOSITION  YI.— THEOREM. 

A  tangent  to  the  ellipse  makes  equal  angles  with  the  two 
.straight  lines  drawn  from  the  point  of  contact  to  the  foci. 

Let  F  and  Ff  be  the  foci  and 
D  any  point  in  the  curve.  Draw 
F'D  and  FD,  and  produce  F'D 
to  //,  making  DH—DF,  and  draw 
FH.  Bisect  FHin  T.  Draw  TD 
and  produce  it  to  t. 

Now,  (by  Cor.  2,  Th.  18,  B.  I,  Geom.),  the  angle  FDT= 
the  angle  HDT,  and  HTD=its  vertical  angle  F'Dl 

Therefore,  FD  T=F'Dt. 

It  now  remains  to  be  shown  that  Tt  meets  the  curve" 
only   at  the  point  D,  and  is,  therefore,  a  tangent. 

If  possible,  let  it  meet  the  curve  in  some  other  point, 
as  t,  and  draw  Ft,  tH,  and  F't. 

(By  Scholium  1,  Th.  18,  B.  I,  Geom.)  Ft=tH. 

To  each  of  these  add  F't; 

Then,  F't-}-  tH=  F't-{-  Ft 


THE    ELLIPSE.  17 

But  F  't  and  tH  are,  together,  greater  than  F1 H,  because 
a  straight  line  is  the  shortest  distance  between  two  points ; 
that  is,  F' t  and  Ft,  the  two  lines  from  the  foci,  are,  together, 
greater  than  FH,  or  greater  than  F'  D+FD;  therefore,  the 
point  t  is  without  the  ellipse,  and  t  is  any  point  in  the  line 
Tt,  except  D.  Therefore,  Tt  is  a  tangent,  touching  the 
ellipse  at  D;  and  it  makes  equal  angles  with  the  lines 
drawn  from  the  point  of  contact  to  the  foci. 

Hence  the  theorem ;  a  tangent,  etc. 

Cor.  The  tangents  at  the  vertices  of  either  axis  are 
perpendicular  to  that  axis ;  and,  as  the  ordinates  are  par 
allel  to  the  tangents,  it  follows  that  all  ordinates  to  either 
axis  must  cut  that  axis  at  right  angles,  and  be  parallel  to 
the  other  axis. 

SCHOLIUM  1. — From  this  proposition  we  derive  the  following 
simple  rule  for  drawing  a  tangent  line  to  an  ellipse  at  any  point : 
Through  the  given  point  draw  a  line  bisecting  tlie  angle  included 
between  the  line  connecting  this  point  with  one  of  the  foci  and  the 
line  produced  connecting  it  with  the  other  focus. 

SCHOLIUM  2.  Any  point  in  the  curve  maybe  considered  as  a  point 
in  a  tangent  to  the  curve  at  that  point. 

It  is  found  by  experiment  that  rays  of  light,  heat  and  sound  are 
incident  upon,  and  reflected  from  surfaces  under  equal  angles ;  that 
is,  for  a  ray  of  either  of  these  principles  the  angles  of  incidence  and 
reflection  are  equal.  Therefore,  if  a  reflecting  surface  be  formed  by 
turning  an  ellipse  about  its  major  axis,  the  light,  heat,  or  sound 
which  proceeds  from  one  of  the  foci  of  this  surface  will  be  concen 
trated  in  the  other  focus. 

Whispering  galleries  are  made  on  this  principle,  and  all  theaters 
and  large  assembly  rooms  should  more  or  less  approximate  this  figure. 
The  concentration  of  the  rays  of  heat  from  one  of  these  points  to 
the  other,  is  the  reason  why  they  are  called  the  foci  or  burning 
points. 

9*  T> 


18  CONIC    SECTIONS. 

PEOPOSITION   YI  I.— THEOREM. 

Tangents  to  the  ellipse,  at  the  vertices  of  a  diameter,  are  par 
allel  to  each  other. 

Let  DD'  be  the  diameter,  and  F' 
and  F  the  foci.  Draw  F'D,  F'D', 
FD,  and  FD . 

Draw  the  tangents,  Tt  and  Ss,  one 
through  the  point  D,  the  other 
through  the  point  D1 .  These  tan 
gents  will  be  parallel. 

By  Cor.  Prop.  5,  F'D'FD  is  a  parallelogram,  and  the 
angle  F'D'F  is  equal  to  its  opposite  angle,  F'DF. 

But  the  sum  of  all  the  angles  that  can  be  made  on  one 
side  of  a  line  is  equal  to  two  right  angles.  Therefore,  by 
leaving  out  the  equal  angles  which  form  the  opposite  an 
gles  of  the  parallelogram,  we  have 

sDfFf+SWF=tDFf+  TDF 

But  (by  Prop.  6)  sD'F'=SD'F-,  and  also  tDF'=  TDF; 
therefore,  the  sum  of  the  two  angles  in  either  member  of 
this  equation  is  double  either  of  the  angles,  and  the  above 
equation  may  be  changed  to 

2SD'F=2tDF'        or        SD'F^tDF' 

But  DF'  and  D'F  are  parallel;  therefore  SD'F  and 
tDF'  are,  in  effect,  alternate  angles,  showing  that  Tt  and 
Ss  are  parallel. 

Cor.  If  tangents  be  drawn  through  the  vertices  of  any 
two  conjugate  diameters,  they  will  form  a  parallelogram 
circumscribing  the  ellipse. 

PROPOSITION  YIII.— THEOREM. 

If,  from  the  vertex  of  any  diameter  of  an  ellipse,  straight 
lines  are  drawn  through  the  foci,  meeting  the  conjugate  diameter, 
the  part  of  either  line  intercepted  by  the  conjugate,  is  equal  to  one 
half  the  major  axis. 


THE    ELLIPSE.  19 

Let  DD'  be  the  diameter,  and  Tt 
the  tangent.  Through  the  center 
draw  EEf  parallel  to  Tt.  Draw  F'D 
and  DF,  and  produce  DF  to  K;  and 
from  F  draw  FG  parallel  to  EEf 
or  Tt. 

2Tow,  by  reason  of  the  parallels,  we  have  the  following 
equations  among  the  angles  : 

tDG=DGF\  A]        (  tDG=DHK 
TDF=DFG  J  ' 


But  (Prop.  6)  WG=  TDF; 

Therefore,  DGF=DFG; 

And,  DHK=DKH 

Hence,  the  triangles  DGF'smd  DHK  are  isosceles. 
Whence,  DG=DF,  and  DH=DK. 

Because  HCia  parallel  to  FG,  and  F'C=CF, 
therefore,  FfH=HG 

Add,  DF=DG 

and  we  have          FfH+DF=DH 

But  the  sum  of  the  lines  in  both  members  of  this  equa 
tion  is  F'D+DF,  which  is  equal  to  the  major  axis  of  the 
ellipse;  therefore,  either  member  is  one  half  the  major 
axis;  that  is,  DH,  and  its  equal,  DK,  are  each  equal  to 
one  half  the  major  axis. 

Hence  the  theorem ;  if  from  the  vertex  of  any  diameter,  etc. 

PROPOSITION    IX.— THEOREM. 

Perpendiculars  from  the  foci  of  an  ellipse  upon  a  tangent, 
meet  the  tangent  in  the  circumference  of  a  circle  whose  diame 
ter  is  the  major  axis. 

Let  F', F  be  the  foci,  G  the  center  of  the  ellipse,  and  D  a 
point  through  which  passes  the  tangent  Tt.  Draw  F'  D 


20  CONIC    SECTIONS. 

and  FD,  produce  F 'D  to  H,  mak-          -  N  H 

ing  DH=FD,  and  produce  FD  to 
G,  making  DG=F'D.  Then  jF'.ff 
and  FG  are  each  equal  to  the  major 
axis,  A' A. 

Draw  FH  meeting  the  tangent  in   A 
T  and  FG  meeting  it  in  t.     Draw 
the  dotted  lines,  CT  and  Ct. 

By  Prop.  6,  the  angle  .FDT^the  angle  F*Dt;  and  since 
opposite  or  vertical  angles  are  equal,  it  follows  that  the  four 
angles  formed  by  the  lines  intersecting  at  D,  are  all  equal. 

The  triangles  DF"  G  and  DHF  are  isosceles  by  con 
struction  ;  and  as  their  vertical  angles  at  D  are  bisected  by 
the  line  Tt,  therefore  F'  t=tG,  FT=  TH,  and  FT  and  F' t 
are  perpendicular  to  the  tangent  Tt. 

Comparing  the  triangles  F'GF  and  F'Ct,  we  find  that 
JFCis  equal  to  the  half  of  F'F,  and  F't,  the  half  of  F'G; 
therefore,  Ct  is  the  half  of  FG- ;  but  A'A=FG;  hence, 
Ct=\A'A=CA. 

Comparing  the  triangles  FF'H  and  FCT9  we  find  the 
sides  FH  and  FFf  cut  proportionally  in  T  and  C; 
therefore,  they  are  equi-angular  and  similar,  and  CT  is 
parallel  to  F'H,  and  equal  to  one  half  of  it.  That  is,  CT 
is  equal  to  CA  ;  and  CA,  CT,  and  Ct  are  all  equal ;  and 
hence  a  circumference  described  from  the  center  (7,  with 
the  radius  CA9  will  pass  through  the  points  T  and  L 

Hence  the  theorem:  perpendiculars  from  the  foci,  etc. 

PROPOSITION    X.— THEOREM. 

The  product  of  the  perpendiculars  from  the  foci  of  an 
dlipse  upon  a  tangent,  is  equal  to  the  square  of  one  half  the 
minor  axis. 

Produce  TC  and  GFf,  and  they  will  meet  in  the  circum 
ference  at  S;  for  FT  and  F't  are  both  perpendicular  to 


THE    ELLIPSE. 


21 


H 


the  same  line  Tt,  they  are  there 
fore  parallel ;  and  the  two  triangles, 
GFTand  CF'S,  having  a  side,  FC, 
of  the  one,  equal  to  the  side,  CF', 
of  the  other,  and  their  angles  equal, 
each  to  each,  are  themselves  equal. 
Therefore,  CS=CT,  S  is  in  the  cir 
cumference,  and  SFf—FT. 

"Now,  since  A' A  and  St  are  two  lines  that  intersect  each 
other  in  a  circle,  therefore  (Th.  17,  B.  3U,  Geom.), 
SFf  x  F't=A'F'  xF'A; 

Or,  FTxF't=A'F'  xF'A. 

But,  by  the  Scholium  to  Prop.  4,  it  is  shown  that 
A'F'  xF'A=  the  square  of  one  half  the  minor  axis. 

Therefore,    FTx  F1 1=  the  square  of  one  half  the  minor 
axis. 

Hence  the  theorem ;  The  product  of  the  perpendiculars,  etc. 

Cor.  The  two  triangles,  FTD  and  F'tD,  are  similar, 
and  from  them  we  have  TF  :  F't=FD  :  DF' ;  that  is, 
perpendiculars  let  fall  from  the  foci  upon  a  tangent,  are  to 
each  other  as  the  distances  of  the  point  of  contact  from  the  foci. 


PROPOSITION    XI.— THEOREM. 

If  a  tangent,  drawn  to  an  ellipse  at  any  point,  be  produced 
until  it  meets  either  axis,  and  from  the  point  of  tangency  an 
ordinate  be  drawn  to  the  same  axis,  one  half  of  the  axis  will  be 
a  mean  proportional  between  the  distances  from  the  center  to 
the  intersections  of  these  lines  with  the  axis. 

Let  Tt  be  a  tangent  at  any 
point  in  the  ellipse,  as  P. 

Draw  F'P  &n&FP,F  and 

F'  being  the  foci,  and  produce  ^~^ —      c        G  F  A 
F'P  to  Q,  making  PQ=PF;  join  T,Q,  and   draw  PG 
perpendicular  to  the  axis  A  A . 


22  CONIC    SECTIONS. 

The  triangles  PFT  and  PTQ  are  equal,  because  PT  is 
common,  PQ=PF  by  construction,  and  the  |__  TPF= 
the  angle  [_  TPQ  (Th.  6). 

Therefore,  TP  bisects  the  angle  FTQ,  and  QT=FT. 

As  the  angle  at  T7  is  bisected  by  TP,  the  sides  about 
tliis  angle  in  the  triangle  F'TQ  are  to  each  other,  as  the 
segments  of  the  third  side,  (Th.  24,  B.  H,  Geom.) 

That  is,  F'  T :  TQ  :  :  F1  P :  PQ 

Or,  F'T  :  FT:  :  F' P  :  PF 

From  this  last  proportion  we  have  (Th.  9,  B.  II,  Geom.), 
F'  T+FT:  F'  T—FT  :  :  F'P+PF :  F'P PF 

Or,  since       F'T+FT=2CT  and  F'P+PF=*2CA, 

by  substitution  we  have 

2CT:  F'F  :  :  2  CM  :  F'P—PF  (1) 

Again,  because  PG  is  drawn  perpendicular  to  the  base 
of  the  triangle  F'PF,  the  base  is  to  the  sum  of  the  two 
sides,  as  the  difference  of  the  sides  is  to  the  difference  of 
the  segments  of  the  base,  (Prop.  6,  PI.  Trig.) 

Whence,  F'F :  F'P+PF :  :  F'P—PF  :  2CG-    (2) 

If  we  multiply  proportions  (1)  and  (2),  term  by  term, 
omitting  in  the  resulting  proportion  the  factor  F'F,  com 
mon  to  the  terms  of  the  first  couplet,  and  the  factor 
F  'P — PF,  common  to  the  terms  of  the  second  couplet, 
we  shall  have 

2CT-.2CA  ::2CA:2CG 

Or,  CT :     CA::    CA  :  CG 

In  like  manner  it  may  be  proved  that 
Ct :  CB  ::  CB-.Cg 

Hence  the  theorem ;  If  a  tangent,  drawn  to  an  ellipse,  etc. 

PROPOSITION    XII.-THEOREM. 

The  sub-tangent  on  either  axis  of  an  ellipse  is  equal  to  the 
corresponding  sub-tangent  of  the  circle  described  on  that  axis  as 
a  diameter. 


THE    ELLIPSE. 

Let  P  be  the  point  of  tan- 
gency  of  the  tangent  line  Tt  to 
the  ellipse,  of  which  A  A'  is  the 
major  axis  and  0  the  center. 
Draw  the  ordinate  P  G  to  this 
axis,  and  produce  it  to  meet  A» c^~  G~ 
the  circumference  of  the  circle  described  on  AA1  as  a 
diameter,  at  J5,  and  draw  EC  and  BT,  T  being  the  inter 
section  of  the  tangent  with  the  major  axis  ;  then  will  the 
line  BT  be  a  tangent  to  the  circumference,  at  the  point  B. 

By  the  preceding  theorem  we  have 

CT  :  OA  :  :  GA  :  CG 

And  since       GA=  CB,  this  proportion  becomes 
CT:  CB:  :  CB  :  CG 

Hence,  the  triangles  CB  T  and  CBG  have  the  common 
angle  (7,  and  the  sides  about  this  angle  proportional ;  they 
are  therefore  similar  (Cor.  2  Th.  17,  B.  II,  Geom.).  But 
CB  G-  is  a  right-angled  triangle;  therefore,  CBT  is  also 
right-angled,  the  right  angle  being  at  B.  Now,  since  the 
line  BT  is  perpendicular  to  the  radius  CB  at  its  extrem 
ity,  it  is  tangent  to  the  circumference,  and  G-T  is  there 
fore  a  common  sub-tangent  to  the  ellipse  and  circle. 

If  a  circumference  be  described  on  the  minor  axis  as 
a  diameter,  it  may  be  proved  in  like  manner  that  the 
corresponding  sub-tangents  of  the  ellipse  and  circle  are 
equal. 

Hence,  the  theorem ;    The  sub-tangent  on  either  axis,  etc. 

SCHOLIUM  1. — This  proposition  furnishes  another  easy  rule  for 
drawing  a  tangent  line  to  an  ellipse,  at  any  point. 

RULE.  On  the  major  axis  as  a  diameter,  describe  a  semi-circum 
ference,  and  from  the  given  point  on  the  ellipse  draw  an  ordinate  to 
the  major  axis;  draw  a  tangent  to  the  semi-circumference  at  the 
point  in  which  the  ordinate  produced  meets  it.  The  line  that  con 
nects  the  point  in  which  this  tangent  intersects  the  major  axis  with  the 
given  point  on  the  ellipse,  will  be  the  required  tangent. 


24  CONIC    SECTIONS. 

SCHOLIUM  2. — Because  CB  T  is  a  right-angled  triangle, 


'=£G  ;  \>M.tA'G-AG=BG* 
Therefore,    CG'GT=A'G'AG 

PROPOSITION    XIII.— THE  OEEM. 

The  square  of  either  semi-axis  of  an  ellipse  is  to  the  square 
of  the  other  semi-axis,  as  the  rectangle  of  any  two  abscissas  of 
the  former  axis  is  to  the  square  of  the  corresponding  ordinate. 

From  any  point,  as  P,  of  the 
ellipse  of  which  C  is  the  center, 
A  A'  the  major,  and  BBf  the 
minor  axis,  draw  the  ordinate 
PG  to  the  major  axis;  then 
it  is  to  be  proved  that 

~CA*  :  ~CB*  :  :  AG'GA  :  PG" 
Through  P  draw  a  tangent  line  intersecting  the  axes 
at  Tand  t ;  then,  by  Prop.  11,  we  have 
CT::  CA::  CA:  CG 


"Whence,  CT>CG=CA 

and  by  multiplying  both  members  of  this  equation  by 
CG,  it  becomes 


dich  may  be  resolved  into  the  proportion 

CA2:CG2::CT:CG    • 
From  this  we  find,  (Cor.  Th.  8,  B.  H,  Geom.), 

ZS2  :  Ol2—  W-2  ::  CT:  GT      (1) 
Again,  drawing  the  ordinate  Pg  to  the  minor  axis,  we 
have 

Ct:  CB::  CB:  Cg  or  PG 
Whence,       Ct  •  PG=CB2 

Multiplying  both  members  of  this  equation  by  PG,  it 
becomes 


THE    ELLIPSE.  25 

a  -  P£2=OB2-P£ 
from  which  we  have  the  proportion 

CB2  :  PG*  :  :  Ct  :  PG 
By  similar  triangles  we  have 

d  :  PG::  GT:  GT 

And,  since  the  first  couplet  in  this  proportion  is  the 
same  as  the  second  couplet  in  the  preceding,  the  terms  of 
the  other  couplets  are  proportional. 

That  is,        W-.PG2::  CT:  GT  (2) 

By  comparing  proportions  (1)  and  (2),  we  obtain 

Cff'.PG'i'.OA2:  CA2—CG2        (3) 

But  CA2—  C(?=(CA+  CG)  (CA—CG)=A'G'AG-, 
Whence,  by  inverting  the  means  in  proportion  (3)  and 
substituting  the  values  of  CA  —  CG  ,  we  have  finally 


or,  CA2  :  W  :  :  AG'AG  :  PG* 

By  a  process  in  all  respects  similar  to  the  above,  we  will 
find  that 


Hence  the  theorem  ;  the  square  of  either  semi-axis,  etc. 

SCHOLIUM  1.  —  From  the  theorem  just  demonstrated  is  readily 
deduced  what  is  called,  in  Analytical  Geometry,  the  equation  of  the 
ellipse  referred  to  its  center  and  axes.  If  we  take  any  point,  as  P9 
on  the  curve,  and  can  find  a  general  relation  between  A  G  and  PG9 
or  between  CG-  and  PG,  the  equation  expressing  such  relation  will 
be  the  equation  of  the  curve.  Let  us  .represent  CA,  one  half  of 
the  major  axis,  by  A}  and  CB,  one  half  of  the  minor  axis,  by  B  ; 
that  is,  the  symbols  A  and  B  denote  the  numerical  values  of  these 
semi-axes,  respectively.  Also,  denote  the  CG  by  x,  and  PG  by  y, 
then  A'G=A-\-x,  and  AG=A—  x-}  and  by  the  theorem  we  have 
J.'  :£2:  :  (A+x)  (A—x)  :/ 

Whence, 

Or, 
3 


CONIC    SECTIONS. 


This  is  the  required  equation  in  which  the  variable  quantities, 
x  and  y,  are  called  the  co-ordinates  of  the  curve,  the  first,  x,  being 
the  abscissa j  and  the  second,  y,  the  ordinate;  the  center  C  from 
which  these  variable  distances  are  estimated,  is  called  the  origin  of 
co-ordinates,  and  the  major  and  minor  axes  are  the  axes  of  co-ordinates. 

Had  we  donoted  A'  G  by  x}  without  changing  y,  then  we  should 
have  AG=2A— x, 

And  J.2  :  B*  :  :  (2 A— «)  x  :  y> 

B* 

Whence,     y*=~(2Ax — x2),  which  is  the  equation  of  the  ellipse 

JO. 

when  the  origin  of  co-ordinates  is  on  the  curve  at  A'.    * 

SCHOLIUM  2. — If  a  circle  be  described  on  either  axis  of  an  ellipse 

as  a  diameter  j  then  any  ordinate  of  the  circle  to  this  axis  is  to  the 

corresponding  ordinate  of  the  ellipse,  as  one  half  of  this  axis  is  to 

one  half  of  the  other  axis. 

Retaining  the  notation  in  Scholium  1,  and  producing  the  ordinate 

PG  to  meet  the  circumference  described  on  A  'A  as  a  diameter,  at 

jP,  we  have,  by  the  theorem, 

A*  :  B2  :  :  (-4+*)  (A—x)  :  y* 


But 

"Whence, 
Or, 
That  is, 


(A+x)  (A—x)  =  GP' 
A*  :£*: 


A    :B:'.GPr    \y 


GP'  :  y  :  :  A  :  B 
By  describing    a  circle  on  BBf  as  a  diameter,  we  may  in  like 
manner  prove  that         pg  :  Pg  :  :  B  :  A 


PROPOSITION  XIV. -THEOREM. 

The  squares  of  the  ordinate  to  either  axis  of  an  ellipse  are 
to  each  other,  as  the  rectangles  of  the  corresponding  abscissas. 
B  Let  AAf  be  the  major,  and  BB' 

the  minor  axis  of  the  ellipse,  and 
jF6r,  P'Gr'  any  two  ordinates  to 
the  first  axis.  Denoting  CGr  by 
by  x,  CG'  by  x',  PG  byy  and 
P'6r'  by  yf,  we  have,  by  Scho.  1, 


THE    ELLIPSE.  27 

Prop.  13,  A*y*  +  B  *x  2=  A2JB2 

and  A*yf*  +  B'2x'*=A*B* 

J32  B* 

Whence,       y*=-(A*—x*)=(A+x)  (A—x)    (1) 


and  y****J&^*)-^A+x?)  (A—xf)  (2) 

Dividing  equation  (1)  by  equation  (2),  member  by  mem 
ber,  and  omitting  the  common  factors  in  the  numerator 
and  denominator  of  the  second  member  of  the  resulting 
equation,  it  becomes 

y\(A+x]  (A-x) 
y'*    (A+x')(A—  xf) 

By  simply  inspecting  the  figure,  we  perceive  that  A+x 
and  A  —  x  represent  the  abscissas  of  the  axis  A  A',  corres 
ponding  to  the  ordinate  y  ;  and  A+x',  and  A  —  x'  those 
corresponding  to  the  ordinate  y'. 

By  placing  the  two  equations  first  written  above,  under 
the  form 


and  proceeding  as  before,  we  should  find 
a?    (£+y)(S-y) 


in  which  B+y,  B — y  are  the  abscessas  of  the  axis 

corresponding  to  the  ordinate  x—CG=Pg',  and 

J3 — y'  are  those  corresponding  to  the  ordinate  #'=  CGf= 

P'g>. 

Hence  the  theorem ;  the  squares  of  the  ordinates,  etc. 

PROPOSITION  XV.— THEOREM. 

The  parameter  of  the  transverse  axis  of  an  ellipse,  or,  the  la- 
tus  rectum,  is  the  double  ordinate  to  this  axis  through  the  focus. 


28  CONIC    SECTIONS. 

Let  F  and  Fr  be  the  foci  of  an 
ellipse  of  which  A  A'  and  BBf  re 
spectively  are  the  major  and  mi- 
nor  axes. 

Through  the  focus  F  draw  the 
double  ordinate  PP'.  Then  will 
PPf  be  the  parameter  of  the  major  axis. 

'  We  will  denote  the  semi-major  axis  by  -A,  the  semi- 
minor  axis  by  JB,  the  ordinate  through  the  focus  by  P,  and 
?ind  the  distance  from  the  center  to  the  focus  by  c. 

The  equation  of  the  curve  referred  to  the  center  and 
axis,  is 


If  in  this  equation  we  substitute  c  for  x,  y  will  become 
P,  and  we  have 


Transposing  the  term  .B2^2,  and  factoring  the  second 
member  of  the  resulting  equation,  it  becomes 

A2P2=B*  (A2-*2)        (l) 

In  the  right-angled  triangle  B  CF,  since  BF=A  (Prop. 
4)  and  Bc=B,  we  have  A2—  c2=jB2. 

Replacing  A2  —  c8  in  eq.  (1)  by  its  value,  that  equation  be 
comes 

A2  -P2=  J32  -B2 
Or,  by  taking  the  square  roots  of  both  members, 

A-P=B-B 

Whence,  A:B::B:P 

Or,  2A:2B::2B:2P 

2P  is  therefore  a  third  proportional  to  the  major  and  mi 
nor  axes,  and  (Def.  14)  it  is  the  parameter  of  the  former 
axis. 
Hence  the  theorem  ;  the  parameter,  etc. 


THE   ELLIPSE. 


29 


PROPOSITION   XVI.— THEOREM. 

The  area  of  an  ellipse  is  a  mean  proportional  between  two 
circles  described,  the  one  on  the  major,  and  the  other  on  the  mi 
nor  axis  as  diameters. 

On  the  major  axis  A  A'  of  the 
ellipse  represented  in  the  figure, 
describe  a  circle,  and  suppose  this 
axis  to  be  divided  into  any  num 
ber  of  equal  parts. 

Through  the  points  of  division 
draw  ordinates  to  the  circle,  and 
join  the  extremities  of  these  consecutive  ordinates,  and 
also  those  of  the  corresponding  ordinates  of  the  ellipse, 
by  straight  lines.  We  shall  thus  form  in  the  semi-circle 
a  number  of  trapezoids,  and  a  like  number  in  the  semi- 
ellipse. 

Let  6r/J,  G'H'  be  two  adjacent  ordinates  of  the  circle, 
and  gH  g'H'  those  of  the  ellipse  answering  to  them ;  and 
let  us  denote  GH  by  F,  G'H'  by  F,  gHbyy,  g'H1  by 
y'j  and  the  part  HHr  of  the  axis  by  x. 

The  trapezoidal  areas,  GHH1  Gf,  gHH'g',  are  respect 
ively  measured  by 

y+  F  y+yr 

—^ x  and^--z  (Th.  34,  B.  I,  Geom.) 

But  (Prop.  13,  Scho.  2) 

A:B::  Y:y 
::  Y>:y' 
Hence  (Th.  7,  B.  II,  Geom.) 


or, 


A-.Bi:  Y+Y-.y+y' 
Y+Y/ 


x 


F+F 

:       2 

y+y' 


If  the  ordinates  following  F,  yf  in  order,  be  represented 
by  F",  #",  etc.,  we  shall  also  have 
3* 


30  CONIC    SECTIONS. 


, 


That  is,  any  trapezoid  in  the  circle  will  be  to  the  cor 
responding  trapezoid  in  the  ellipse,  constantly  in  the  ratio 
of  A  to  jB;  and  therefore  the  sum  of  the  trapezoids  in  the 
circle  will  be  to  the  sum  of  the  trapezoids  in  the  ellipse 
as  A  is  to  B;  and  this  will  hold  true,  however  great  the 
number  of  trapezoids  in  each. 

Calling  the  first  sum  S,  and  the  second  s,  we  shall  then 
have 

A:B::S:s 

But,  when  the  number  of  equal  parts  into  which  the 
axis  AAf  is  divided,  is  increased  without  limit,  S  becomes 
the  area  of  the  semi-circle  and  s  that  of  the  semi-ellipse. 

Therefore,    A  :  B  : :  area  semi-circle  :  area  semi-ellipse. 

Or,  A  :  B  : :  area  circle  :  area  ellipse. 

By  substituting  in  this  last  proportion  for  area  circle,  its 
value  xA2,  it  becomes 

A  :  B  : :  xA2 :  area  ellipse. 

"Whence  area  ellipse=7rJ..B, 

which  is  a  mean  proportional  between  xA2  and  xB2. 

Hence  the  theorem ;  the  area  of  an  ellipse,  etc. 

SCHOLIUM. — This  theorem  leads  to  the  following  rule  in  mensu 
ration  for  finding  the  area  of  an  ellipse. 

~Ruii'E.=Multipli/  the  product  of  the  semi-major  and  semi-minor 
axes  by  3.1416. 

PROPOSITION   XVII.— THEOREM. 

If  a  cone  be  cut  by  a  plane  making  an  angle  with  the  base  less 
than  that  made  by  an  element  of  the  cone,  the  section  is  an  el 
lipse. 

Let  VloQ  the  vertex  of  a  cone,  and  suppose  it  to  be  cut 
by  a  plane  at  right-angles  to  the  plane  of  the  opposite 


THE    ELLIPSE.  31 

elements,  VN  VB,  these  elements 
being  cut  by  the  first  plane  at  A 
and  B.  Then,  if  the  secant  plane 
be  not  parallel  to  the  base  of  the 
cone,  the  section  will  be  an  ellipse, 
of  which  AB  is  the  major  axis. 

Through  any  two  points,  F  and 
H,  on  ABy  draw  the  lines  KL,  MN, 
parallel  to  the  base  of  the  cone,  and 
through  these  lines  conceive  planes  to  be  passed  also  par 
allel  to  this  base.  The  sections  of  the  cone  made  by  these 
planes  will  be  circles,  of  which  KGL  and  MIN  are  the 
semi-circumferences,  passing  the  first  through  6r,  and  the 
second  through  J,  the  extremities  of  the  perpendiculars 
to  BAj  lying  in  the  section  made  by  the  oblique  plane. 

The  triangles  AFL,  AHN,  are  similar  ;  so  also  are  the 
triangles  BMH,  BKF;  and  from  them  we  derive  the  fol 
lowing  proportions  : 

AF-.FLr.AHiHN 
BF:KF::BH:HM 

By  multiplication,  AF-BF:  FL-KF:  :  AH-BH:  UN-  JIM 
Because  KL  is  a  diameter  of  a  circle,  and  FG  an  ordi- 
nate  to  this  diameter,  we  have 


and  for  a  like  reason, 

Therefore,     AF-BF  :  FG2  :  :  AH-HB  :  Iff 

or,          AF-BF  :  AH-HB  :  :  FG2  :  HP 

This  proportion  expresses  the  property  of  the  ellipse 
proved  in  (Prop.  14)  ;  and  the  section  A  GIB  is,  therefore, 
an  ellipse. 

Hence  the  theorem  ;  if  a  cone  be  cut,  etc. 

SCHOLIUM.  —  The  proportion  AF-  BF  :  AH-HB::FG?  :  Hf 
would  still  hold  true,  were  the  line  AB  parallel  to  the  base  of  the 
cone,  and  the  section  a  circle  ;  the  ratios  would  then  become  equal 


32 


CONIC    SECTIONS. 


to  unity.     The  circle  may  therefore  be  regarded  as  a  particular  case 
of  the  ellipse.  Jj 


PROPOSITION  XYIII.— THEOREM. 

If,  from  one  of  the  vertices  of  each  of  two  conjugate  diameters 
of  an  ellipse,  ordinates  be  drawn  to  either  axis,  the  sum  of  the 
squares  of  these  ordinates  will  be  equal  to  the  square  of  the 
other  semi-axis. 


an  ellipse,  of  which 
A  A'  is  the  major  and 
BBr  the  minor  axis ; 
also  let  P§,  P'g'be 
any  two  conjugate 
diameters.  Through 
the  vertices  of  these 
diameters  draw  the  tangents  to  the  ellipse  and  the  ordi 
nates  to  the  axes,  as  represented  in  the  figure.  Then  we 
are  to  prove  that 


and  CB=(PG)2+(P'G'}2=(Cgf+(OgJ 

Now  (by  Prop.  11)  we  have 

GT:  CA::  CA  :  OG, 

also,  Of  :  CA::  CA:  On 

"Whence, 

and 

Therefore, 

which,  resolved  into  a  proportion,  gives 

Of  :  CT::  OG  :  Cn  (2) 

By  the  construction,  it  is  evident  that  the  triangles 
OPT,  CQ't,  are  similar,  as  are  also  the  triangles  PGT 
and  CQn. 


THE    ELLIPSE. 


33 


Prom  these  triangles  we  derive  the  proportions 

Ct'  :  CT:  :  CQf  :  PT 
CQ!  :PT:  :  On    :  GT 
Whence,  Ct  :  CT  :  :  On     i  GT 

Comparing  the    last  proportion  with  proportion   (2) 
above,  we  have 

CG:  Cn::  Cn:  GT 
Whence,  (Crif=CG'GT 

But        GT=  CT—CG;  then  (Cnf=  CG  (CT—CG), 
from  which  we  get 

(Cn)2+  CG*=  CG-  CT=  CA*     (See  eq.  1.) 
Substituting,  in  this  equation,  for  (<7ft)2,  its  equal  CGr  > 
it  becomes 


In  a  similar  manner  it  may  be  proved  that 


Hence  the  theorem  ;  if  from  one  of  the  vertices  of  each,  etc. 


PROPOSITION    XIX.— THEOEEM. 

The  sum  of  the  squares  of  any  two  conjugate  diameters  of 
an  ellipse  is  a  constant  quantity,  and  equal  to  the  sum  of  the 
squares  of  the  axes. 

The  annexed  fig 
ure,  being  the  same 
as  that  employed  in  ^ 
the  preceding  prop 
osition,  by  that  prop 
osition  we  have 


CA=CG  +  CG' 


and 

By  addition,  ~CA9 


CG*+  PG2+  CG' 


34 


CONIC    SECTIONS. 


But  CG  and  PG  are  the  two  sides  of  the  right-angled 
triangle  CPG,  and  CG'  and  PfGr  are  the  two  sides  of 
the  right-angled  triangle  CPf  Gr ; 

Therefore,          OA.2  +  ~CB* 

Whence,          4CA2+4tCB* 

The  first  member  of  this  equation  expresses  the  sum  of 
the  squares  of  the  axes,  and  the  second  member  the  sum 
of  the  squares  of  the  two  conjugate  diameters. 

Hence  the  theorem ;  the  sum  of  the  squares  of  any  two,  etc. 


PROPOSITION    XX.— THEOREM. 

The  parallelogram  formed  by  drawing  tangents  through  the 
vertices  of  any  two  conjugate  diameters  of  an  ellipse,  is  equal  to 
the  rectangle  of  the  axes. 

Employing  the 
figure  of  the  last 
two  propositions,  we 
have,  from  proposi 
tion  18, 


from  which,  by  trans-  o 

position  and  factoring  the  second  member,  we  get 

gG2=(CA+CG')  (CA—CGf)=AG''A'Gf 
But          CA2 :  CB2  ::  A_G'-A'G^_PfG'2;     (Prop.  13.) 

CG2:    PfG'Z 
CG  :    P'G'^Qn  (1) 
CA   :    CG    (2)  (Prop.  11.) 


CB2 

CB 

CA 


Whence,  C 
Or,  CA 

But,          CT 

Multiplying  proportions  (1)  and  (2),  term  by  term, 
omitting,  in  the  first  couplet  of  the  resulting  proportion, 
the  common  factor  CA,  and  in  the  second  couplet  the 
common  factor  CG,  we  find 

CT:  CB::  CA:  Qfn 


THE    ELLIPSE.  35 

Whence,          CT-  Q  fn=  CA  -  CB 
Or,  4CT-Q'n=4£i-CB 

The  first  member  of  this  equation  measures  eight  times 
the  area  of  the  triangle  CQ'  T,  and  this  triangle  is  equiva 
lent  to  one  half  of  the  parallelogram  CQ'mP,  because  it 
has  the  same  base,  CQ ',  as  the  parallelogram,  and  its  vertex 
is"  in  the  side  opposite  the  base.  This  parallelogram  is 
obviously  one  fourth  of  that  formed  by  the  tangent 
lines  through  the  vertices  of  the  conjugate  diameters; 
4CT.Q'n  therefore,  measures  the  area  of  this  parallelo 
gram.  Also,  4  CA-CB  is  the  measure  of  the  rectangle  that 
would  be  formed  by  drawing  tangent  lines  through  the 
vertices  of  the  major  and  minor  axes  of  the  ellipse. 

Hence,  the  theorem ;  the  parallelogram  formed,  etc. 

PROPOSITION    XXI.-THEOREM. 

If  a  normal  line  be  drawn  to  an  ellipse  at  any  point,  and] 
also  an  ordinate  to  the  major  axis  from,  the  same  point,  then 
will  the  square  of  the  semi-major  axis  be  to  the  square  of  the 
semi-minor  axis,  as  the  distance  from  the  center  to  the  foot  of  \ 
the  ordinate  is  to  the  sub-normal  on  the  major  axis. 

Let  P  be  the  assumed  point 
in  the  ellipse,  and  through  this 
point  draw  the  tangent  P  I7,  the 
normal  PD,  and  the  ordinate 
PG,  to  the  major  axis ;  then  C 
being  the  center  of  the  ellipse, 
and  A  denoting  the  semi-major,  and  B  the  semi-minor 
axis,  it  is  to  be  proved  that 

A2 :  B2 : :  CG  :  DG 
By  (Prop.  13)  we  have 

A2:B2::A'G-AG:TG*  (1) 

and  because  DPT  is  a  right-angled  triangle,  and  PGr  is  a 


36 


CONIC    SECTIONS. 


perpendicular  let  fall  from  the 'vertex  of  the  right-angle 
upon  the  hypotenuse,  we  also  have 

(Th.  25,  B.  II,  Geom.)         ~PG?=DG'GT 
But  A'  G- A  G=  CG-  G  T  (Scho.  2,  Prop.  12) 

Substituting  in  proportion  (1),  for  the  terms  of  the  sec 
ond  couplet,  their  values,  it  becomes 

A2:£2::  CG'GT-.DG'GT 
or  A2:£2::CG:DG. 

Hence  the  theorem  ;  if  a  normal  line  be  drawn,  etc. 
Cor.     If  CG—Xj  then  this  theorem  will  give  for  the 

IP 

A2 
pression. 


/i 

mibnormal,D6r,  the  value  —  x,  which  is  its  analytical  ex- 


ST 


PROPOSITION    XXII.— THE  OEEM. 

If  two  tangents  be  drawn  to  an  ellipse,  the  one  through  the 
vertex  of  the  major  axis  and  the  other  through  the  vertex  of  any 
other  diameter,  each  meeting  the  diameter  of  the  other  produced, 
the  two  tangential  triangles  thus  formed  will  be  equivalent. 

Let  PPf  be  any  diameter  of 
the  ellipse  whose  major  axis 
is  AAf.     Draw  the  tangents 
JJVand  PT,  the  first  meeting 
the  diameter  produced  at 
and  the  second  the  axis  pro- 
duced  at  T;  the  triangles  CAN  and  CPT  thus  formed  are 
equivalent. 

Draw  the  ordinate  PD;  then  by  similar  triangles  we 
have 

CD:  CM::  CP:  CN 

But  CD  :  CA  ::CA:  CT  (Prop.  11) 

Whence      CP:  CN: :  CA  :  CT 

Therefore,      CP-  CT=  CN-  CA 


THE    ELLIPSE.  37 

Multiplying  both  members  of  this  equation  by  sin.  Cy 
it  becomes 

CP-  CT  sin.  a=  CN-  CA  sin.  0 

or,  iCT'CP&m.C^CA'CNsm.C        (1) 

But         CP-  sin.  C=PD,  and  CN-  sin.  C=AN; 
therefore  the  first  member  of  equation  (1)  measures  the 
area  of  the  triangle  CPT,  and  the  the  second  member 
measures  that  of  the  triangle  CAN. 

Hence  the  theorem ;  if  two  tangents  be  drawn  to  an,  etc. 

Cor.  1.     Taking  the  common  area  CAJEP,  from  each 
triangle,  and  there  is  left  &PEN=&AET. 

Cor.  2.     Taking  the  common  A  CDP,  from  each  trian 
gle,  and  there  is  left  AP-DT=  trapezoidal  area  PDAN. 

PROPOSITION   XXIII.-THEOREM. 

The  supposition  of  Proposition  22  being  retained,  then,  if  a 
secant  line  be  drawn  parallel  to  the  second  tangent,  and  ordi- 
nates  to  the  major  axis  be  drawn  from  the  points  of  intersec 
tion  of  the  secant  with  the  curve,  thus  forming  two  other  tri 
angles,  these  triangles  will  be  equivalent  each  to  each  to  the  cor- 
responding  trapezoids  cut  off,  by  the  ordinates,  from  the  trian 
gle  determined  by  the  tangent  through  the  vertex  of  the  major  axis. 
» 

Draw  the  secant  QnS  par 
allel  to  the  tangent  PT,  and 
also  the  ordinates  QJR,  ng,  pro 
ducing  the  latter  to  p.  Then  A'^  |n'  //(:S\  ^  ^A  "ST 
is  A£§j£=trapezoid  ANVJR, 
and  A£%=trapezoid  ANpg.  V' • 

The  three  triangles,  CVE,CPD,CNA  are  similar,  by 
construction ;  therefore, 

&CNA  :  AOFD  :  :  CM2 :  :  ~CP* 
"Whence, 

trapezoid  ANPD  :  &CNA : :  ~CA*—~C~ff :  GZ2(1) 
(Th.  8,  B.  II,  Geom.) 
4 


38  CONIC    SECTIONS. 

In  like  manner, 

trapezoid  ANVR  :  &CNA  :  :  CA'—CR2  :  ~CA2  (2) 
Dividing  proportion  (1)  by  (2),  term  by  term,  we  get 

trapezoid  ANPD       m   ~CA* 


trapezoid  ANVR  '       '  ~(JA2  _ 
Whence, 
trapez.  ANPD  rjrapez.  ANVE  :  :  'CA2—UT>2  :  CAZ—CR2 

But    JPD2  :  ~QR2  :  :  A'D'DA  :  A'R-RA,  (Prop.  14)  ; 
and  since 

A'D=*  CA+  CD,  A'R=  CA+  CR,  DA=CA—CD  and 
RA=CA—CR,  we  have 

~PD2  :  QlR2  :  :  (CA+  CD)  (CA—CD)  :  (CA+  CR) 

(CA—CE)::  ~CA2—Clf  :  ~CA2—CIl2 
Therefore, 

trapezoid  ANPD  :  trapezoid  ANVR  :  :~PD2  :  ~QR2, 
But  the  trapezoid  ANPD=&TPD,  (Cor.  2,  Prop.  22); 
whence, 


;2 


A  TPD  :  trapezoid  ANVR  ::PD.::  QR      (3) 
and  since  the  triangles   TPD  and  SQR  are  similar,  we 
have 

ATPZ) :  ASQR  :  :  ~Plf  :  'QR2    (4) 
By  comparing  proportions  (3)  and  (4)  we  find 

A  TPD  :   trapezoid  ANVR  :  :  &TPD  :  &SQR 
"Whence,        trapezoid  ANVR=&SQR; 
and  by  a  similar  process  we  should  find  that 

trapezoid  ANpg=A.Sng. 

Hence  the  theorem ;  if  a  secant  line  be  drawn  parallel,  etc. 
Cor.  1.     Taking  the  trapezoid  ANpg  from  the  trapezoid- 
ANVR,  and  the  A£%  from  the  &SQR,  we  have 

trapezoid  gpVR= trapezoid  gnQR. 
Cor.  2.  The  spaces  ANVR,  TPVR,  and  SQR  are  equiv 
alent,  one  to  another. 

Cor.  3.     Conceive  QR  and  QS  to  move  parallel  to  their 
present  positions,  until  R  coincides  with  C;  then  QR 


THE    ELLIPSE. 


becomes  the  semi-minor  axis,  the  space  ANVE  the  tri 
angle  ANC,  and  the  &QKS  equivalent  to  the  ACPI7. 

PROPOSITION  XX  I  Y. -THEOREM. 

Any  diameter  of  the  ellipse  bisects  all  of  the  chords  of  the  el 
lipse  drawn  parallel  to  the  tangent  through  the  vertex  of  the 
diameter. 


*£ 


A  ST 


By  Cor.  1  to  the  preceding 
proposition  we  have 

If  from  each  of  these  equals 

we  subtract  the  common  area 

gnm  VR,  there  will  remain  the 

Aranp,  equivalent  to  the  AQw  V;  and  as  these  triangles 

are  also  equi-angular,  they  are  absolutely  equal. 

Therefore,  Qm—mn. 

Hence  the  theorem ;  any  diameter  of  the  ellipse  bisects,  etc. 

REMARK. — The  property  of  the  ellipse"  demonstrated  in  this 
proposition  is  merely  a  generalization  of  that  previously  proved  in 
Prop.  3. 


PROPOSITION  XXV.— THEOREM. 

The  square  of  any  semi-diameter  of  an  ellipse  is  to  the  square 
of  its  semi-conjugate,  as  the  rectangle  of  any  two  abscissas  of 
the  former  diameter  is  to  the  square  of  the  corresponding  ordi- 
nate. 

Let  A  A'  be  the  major  axis 
of  the  ellipse,  CP  any  semi- 
diameter  and  CP  its  semi- 
conjugate.  Draw  the  tan- 
gents  TP  and  AN,  the  ordi- 
nate  Qm,  producing  it  to  meet 
the  axis  at  S;  and  Pf  V,  parallel  to  AN,  and  in  other 


40  CONIC    SECTIONS. 

respects  make  the  construction  as  indicated  in  the  figure. 
It  is  then  to  be  proved  that 

OP2 :  OP2 :  :  Pm-mP' :  Qm 

"Now  in  the  present  construction,  the  triangles  CPU' 
and  CV'R'  take  the  place  of  the  triangles  SQR  and  CVR 
respectively,  in  Prop.  23 ;  and  hence  by  that  proposition, 
the  triangles  CP'  V,  CAN,  and  CPT  are  equivalent  one 
to  a,nother. 

The  triangles  CPT  and  CmS  are  similar ;  therefore, 


"Whence, 


ACtotf :  :  CP2  :  Cm 


AGP77:  ^CPT—ACmS:  :  CP2  :  CP*—Cm 

Or,     A  OPT7:  trapez.  mPTS  :  :  ~CP2  :  CP2—~Cin   <& 
From  the  similar  triangles,  CPr  V  and  mQV,  we  have 

A  OF  V  :  Aw§F  :  :  OF2  :  m~Q2 
But  area       Sm  VR+  A  CVJR+  Am  Q  F=  area  Sm  VR+ 

A  <7F-£-f  trapez.  mPTS,  (Prop.  23.)  ;  therefore,  Aw§F= 

trapez.  mPTS  ;  also  A<7P;  V'=&CPT. 

Substituting  these  values  in  the  preceding  proportion, 

it  becomes 

ACP!T  :  trapez.  mPTS  :  :  OP2  :  m§2   (2) 
By  comparing  proportions  (1)  and  (2),  we  get 

CP2  :  OP2—  ~Cm  :  :  CP2  :  ^Q2 
Or,  CP2  :  CPf2  :  :  ~CP2—Cm  :  m<? 

"Whence,   OP2  :  OF2  :  :  (CP+Cm)  (CP—Cm)  :  ^Q2 


Or,  CP2  :  OF2  :  :  P'm-mP  : 

Hence  the  theorem  ;  the  square  of  any  semi-diameter  ',  etc. 

REMARK.  The  property  of  the  ellipse  relating  to  conjugate 
diameters,  established  by  this  proposition,  is  but  the  generalization 
of  that  before  demonstrated  in  reference  to  the  axes,  in  Prop.  13. 


THE  PAEABOLA.  41 


THE  PARABOLA. 


DEFINITIONS. 

1.  The  Parabola  is  a  plane  curve,  generated  by  the 
motion  of  a  point  subjected  to  the  condition  that   its 
distances  from  a  fixed  point  and  a  fixed  straight  line  shall 
be  constantly  equal. 

2.  The  fixed  point  is  called  the 
focus  of  the  parabola,  and  the  fixed 
line  the  directrix. 

Thus,  in  the  figure,  Fis  the  focus 
and  BB"  the  directrix  of  the  para 
bola  PFP'P",  etc. 

3.  A  Diameter  of  the  parabola  is  a  line  drawn  through 
any  point  of  the  curve,  in  a  direction  from  the  directrix, 
and  at  right-angles  to  it. 

4.  The  Vertex  of  a  diameter  is  the  point  of  the  curve 
through  which  the  diameter  is  drawn. 

5.  The  Principal  Diameter,  or  the  Axis,  of  the  parabola 
is  the  diameter  passing  through  the  focus.     The  vertex  of 
the  axis  is  called  the  principal  vertex,  or  simply  the  vertex 
of  the  parabola. 

The  vertex  of  the  parabola  bisects  the  perpendicular 
distance  from  the  focus  to  the  directrix,  and  all  the  diam 
eters  of  the  parabola  are  parallel  lines. 

6.  An  Ordinate  to  a  diameter  is  a  straight  line  drawn 
from  any  point  of  the  curve  to  the  diameter,  parallel  to  the 

4* 


42 


CONIC    SECTIONS. 


tangent  line  through  its  vertex.  Thus, 
PD,  drawn  parallel  to  the  tangent  V  T, 
is  an  ordinate  to  the  diameter  VT>.  It 
will  he  shown  that  DP=.DGr;  and  hence 
PGr  is  called  a  double  ordinate. 

7.  An  Abscissa  is  the  part  of  the  diam 
eter  hetween  the  vertex  and  an  ordinate. 
Thus,  VfD  is  the  ahscissa  corresponding 
to  the  ordinate  PD. 

8.  The  Parameter  of  any  diameter  of  the  parahola  is 
one  of  the  extremes  of  a  proportion,  of  which  any  ordi 
nate  to  the  diameter  is  the  mean,  and  the  corresponding 
abscissa  the  other  extreme. 

9.  The  parameter  of  the  axis  of  the  parahola  is  called 
the    principal  parameter,  or  simply  the  parameter  of  the 
parabola.     It  will  be   shown  to  be  equal  to  the  double 
ordinate  to  the  axis  through  the  focus.     Thus,  BBf,  .the 
chord  drawn  through  the  focus  at  right-angles  to  the  axis, 
is  the  parameter  of  the  parabola. 

The  principal  parameter  is  sometimes  called  the  latus- 
rectum. 

10.  A  Sub-tangent,  on  any  diameter,  is  the  distance  from 
the  point  of  intersection  of  a  tangent  line  with  the  diameter 
produced  to  the  foot  of  that  ordinate  to  this  diameter  that 
is  drawn  from  the  point  of  contact. 

11.  A  Sub-normal,   on   any   diameter,  is 
the  part  of  the  diameter  intercepted  be 
tween  the  normal  to  the  curve,  at  any  point, 
and  the  ordinate  from  the  same  point  to 
the  diameter.     Thus,  in  the  figure,   V'N 
being  any  diameter,  PT  a  tangent,  and 
PN  a  normal  at  the  point  P,  and  PQ  an 

ordinate  to  the  diameter;  then  TQ  is"  a  sub-tangent  and 
QN&  sub-normal  on  this  diameter. 


THE    PARABOLA.  43 

"When  the  terms,  sub-tangent  and  sub-normal,  are  used 
without  reference  to  the  diameter  on  which  they  are  ta 
ken,  the  axis  will  always  be  understood. 

PROPOSITION   I.-PROBLEM. 

To  describe  a  parabola  mechanically. 

Let  CD  be  the  given  line,  and  F  the 
given  point.     Take  a  square,  as  DBG, 
and  to  one  side  of  it,  GB,  attach  a  thread,   B 
and  let  the  thread  be  of  the  same  length  31 
as  the  side  GB  of  the  square.    Fasten  one   c 
end  of  the  thread  at  the  point  G,  the  other  end  at  F. 

Put  the  other  side  of  the  square  against  the  given  line, 
CD,  and  with  the  point  of  a  pencil,  in  the  thread,  bring 
the  thread  up  to  the  side  of  the  square.  Slide  the  side 
BD  of  the  square  along  the  line  CD,  and  at  the  same  time 
keep  the  thread  close  against  the  other  side,  permitting 
the  thread  to  slide  round  the  point  of  the  pencil.  As  the 
side  BD  of  the  square  is  moved  along  the  line  CD,  the 
pencil  will  describe  the  curve  represented  as  passing 
through  the  points  Fand  P. 

For         <7P+P^=the  length  of  the  thread, 

and         GP+PB=ihe  length  of  the  thread. 

By  subtraction,  PF—PB=0,  or  PF=*PB. 

This  result  is  true  at  any  and  every  position  of  the 
point  P;  that  is,  it  is  true  for  every  point  on  the  curve 
corresponding  to  definition  1. 

Hence,  FV=  VH. 

If  the  square  be  turned  over  and  moved  in  the  opposite 
direction,  the  other  part  of  the  parabola,  on  the  other  side 
of  the  line  FH,  may  be  described. 

Cor.  It  is  obvious  that  chords  of  the  curve  which  are 
perpendicular  to  the  axis,  are  bisected  by  it. 


44  CONIC    SECTIONS. 

PROPOSITION  II.— THEOREM. 

Any  point  within  the  parabola,  or  on  the  concave  side  of 
the  curve,  is  nearer  to  the  focus  than  to  the  directrix;  and  any 
point  without  the  parabola,  or  on  the  convex  side  of  the  curve, 
is  nearer  to  the  directrix  than  to  the  focus. 

Let  jPbe  the  focus  and  HBf  the  directrix  B' 
of  a  parabola. 

First. — Take  A,  any  point  within  the  curve. 
From  A  draw  AFio  the  focus,  and  AB  per-  B 
pendicular  to  the  directrix;  then  will  AF 
be  less  than  AB. 

Since  A  is  within  the  curve,  and  B  is  without  it,  the 
line  AB  must  cut  the  curve  at  some  point,  as  P.     Draw 
PF.     By  the  definition  of  the  parabola,  PB=  PF;  adding 
PA  to  each  member  of  this  equation,  we  have 
PB+PA=BA=PA+PF 

But  PA  and  PF  being  two  sides  of  the  triangle  APF, 
are  together  greater  than  the  third  side  AF;  therefore 
their  equal,  BA,  is  greater  than  AF. 

Second. — 'Now  let  us  take  any  point,  as  A',  without  the 
curve,  and  from  this  point  draw  A'F  to  the  focus,  and 
A'Br  perpendicular  to  the  directrix. 

Because  A'  is  without  the  curve  and  F  is  within  it, 
AF  must  cut  the  curve  at  some  point,  as  P.  From  this 
point  let  fall  the 'perpendicular,  BP,  upon  the  directrix, 
and  draw  A  B. 

As  before,  PB=PF;  adding  A'P  to  each  member  of 
this  equation,  and  we  have  A'P+PB=A'P+PF=A'F. 
But  A'P  and  PB  being  two  sides  of  the  triangle  A'PB, 
are  together  greater  than  the  third  side,  A'  B ;  therefore 
their  equal,  A'F,  is  greater  than  A'B.  Now  A'B,  the  hy 
potenuse  of  the  right-angled  triangle  A'BB'  is  greater 
than  either  side;  hence,  A'B  is  greater  than  A'B' ;  much 
more  then  is  A'F  greater  than  A'B'. 

Hence  the  theorem;  any  point  within  the  parabola,  etc. 


THE    PAEABOLA.  45 

Cor.  Conversely:  If  the  distance  of  any  point  from  the 
directrix  is  less  than  the  distance  from  the  same  point  to  the  fo 
cus,  such  point  is  without  the  parabola;  and,  if  the  distance 
from  any  point  to  the  directrix  is  greater  than  the  distance  from 
the  same  point  to  the  focus,  such  point  is  within  the  parabola. 

First.— Let  A'  be  a  point  so  taken  that  A'B'<A'F. 
Now  A'  is  not  a  point  on  the  curve,  since  the  distances 
A'B'  and  AfF  are  unequal;  and  Ar  is  not  within  the 
curve,  for  in  that  case  A'B'  would  be  greater  than  A'F 
according  to  the  proposition,  which  is  contrary  to  the  hy 
pothesis.  Therefore  A'  being  neither  on  nor  within  the 
parabola,  must  be  without  it. 

Second. — Let  A  be  a  point  so  taken  that  AB>AF. 
Then,  as  before,  A  is  not  on  the  curve,  since  AF  and  AB 
are  unequal ;  and  A  is  not  without  the  curve,  for  in  that 
case  AB  would  be  less  than  AF,  which  is  contrary  to  the 
hypothesis.  Therefore,  since  A  is  neither  on  nor  without 
the  parabola,  it  must  be  within  it. 

PROPOSITION  III.— THEOREM. 

If  a  line  be  drawn  from  the  focus  of  a  parabola  to  any  point 
of  the  directrix,  the  perpendicular  that  bisects  this  line  will  be  a 
tangent  to  the  curve. 

Let  F  be  the  focus,  and  HD  the  di 
rectrix  of  a  parabola. 

Assume  any  point  whatever,  as  B,  in  B 
the  directrix,  and  join  this  point  to  the 
focus  by  the  line  BF;  then  will  tA,  the  UF  v  F 
perpendicular  to  BF  through  its  middle  point  t,  be  a  tan 
gent  to  the  parabola.  Through  B  draw  BL  perpendicu 
lar  to  the  directrix,  and  join  P,  its  intersection  with  tP, 
to  the  focus.  Then,  since  P  is  a  point  in  the  perpendic 
ular  to  BF  at  its  middle  point,  it  is  equally  distant  from 
the  extremities  of  BF;  that  is,  PB=PF.  P  is  there- 


46  CONIC    SECTIONS. 

fore  a  point  in  the  parabola,  (Def.  1).     Hence,  the  line  tP 
meets  the  curve  at  the  point  P. 

"We  will  now  prove  that  all  other  points  in  the  line  tP 
are  without  the  parahola.  Take  A9  any  point  except  P 
in  the  line  tP,  and  draw  AF,  AB;  also  draw  AD  perpen 
dicular  to  the  directrix.  AF  is  equal  to  AB,  because  A 
is  a  point  in  the  perpendicular  to  BF  at  its  middle  point; 
but  AB,  the  hypotenuse  of  the  right-angled  triangle  ABD, 
is  greater  than  the  side  AD;  therefore  AD  is  less  than 
AF,  and  the  point  A  is  without  the  parabola.  (Cor., 
Prop.  2).  The  line  tA  and  the  parabola  have  then  no 
point  in  common  except  the  point  P.  This  line  is  there 
fore  tangent  to  the  parabola. 

SCHOLIUM  1. — The  triangles  BPt  and  FPt  are  equal;  therefore 
the  angles  FPt  and  BPt  are  equal.  Hence,  to  draw  a  tangent  to 
the  parabola  at  a  given  point,  we  have  the  following 

RULE. — From  the  given  point  draw  a  line  to  the  focus,  and  an 
other  perpendicular  to  the  directrix,  and  through  the  given  point 
draw  a  line  bisecting  the  angle  formed  by  these  two  lines.  The  bi 
secting  line  will  be  the  required  tangent. 

SCHOLIUM  2. — Just  at  the  point  Pthe  tangent  and  the  curve  co 
incide  with  each  other ;  and  the  same  is  true  at  every  point  of  the 
curve.  Now,  because  the  angles  BPt  and  FPt  are  equal,  and 
the  angles  BPt  and  LPA  are  vertical,  it  follows  that  the  angles 
LPA  and  FPt  are  equal.  Hence  it  follows,  from  the  law  of  re 
flection,  that  if  rays  of  light  parallel  to  the  axis  VF  be  incident 
upon  the  curve,  they  will  all  be  reflected  to  the  focus  F.  If  there 
fore  a  reflecting  surface  were  formed,  by  turning  a  parabola  about 
its  axis,  all  the  rays  of  light  that  meet  it  parallel  with  the  axis,  will 
be  reflected  to  the  focus ;  and  for  this  reason  many  attempts  have 
been  made  to  form  perfect  parabolic  mirrors  for  reflecting  telescopes. 

If  a  light  be  placed  at  the  focus  of  such  a  mirror,  it  will  reflect 
all  its  rays  in  one  direction ;  hence,  in  certain  situations,  parabolic 
mirrors  have  been  made  for  lighthouses,  for  the  purpose  of  throwing 
all  the  light  seaward. 

Cor.  1.  The  angle  BPF  continually  increases,  as  the 


THE    PABABOLA.  47 

pencil  P  moves  toward  "F,  and  at  V  it  becomes  equal  to 
two  right  angles  ;  and  the  tangent  at  V  is  perpendicular 
to  the  axis,  which  is  called  the  vertical  tangent. 

Cor.  2.  The  vertical  tangent  bisects  all  the  lines  drawn  from 
the  focus  of  a  parabola  to  the  directrix. 

Let  Vt  be  the  vertical  tangent ;  then  because  the  two 
right-angled  triangles  FVt  and  FHB  are  similar,  and 
VF=  VH,  we  have  Ft=tB. 


PROPOSITION     I  V.— THEOREM. 

The  distance  from  the  focus  of  a  parabola  to  the  point 
of  contact  of  any  tangent  line  to  the  curve,  is  equal  to  the  dis 
tance  from  the  focus  to  the  intersection  of  the  tangent  with  the 
axis. 

Through  the  point  P  of  the  parabola 
of  which  F  is  the  focus  and  BH  the 
directrix,   draw  the  tangent  line   PT, 
meeting  the  axis  produced  at  the  point  f  k  v  ir  i 
T;  then  will  FP  be  equal  to  FT 

Draw  PB  perpendicular  to  the  directrix,  and  join  F,B. 

The  angles  BPT  and  TPF  are  equal,  (Scho.  1,  Prop.  3) ; 
and  since  PB  is  parallel  to  TG,  the  alternate  angles  BP  T, 
and  PTC  are  also  equal.  Hence  the  angle  TPF  is  equal 
to  the  angle  PTF,  and  the  triangle  PFT  is  isosceles; 
therefore  FP=FT. 

Hence  the  theorem ;  the  distance  from  the  focus  to,  etc.    ! 

SCHOLIUM. — To  draw  a  tangent  line  to  a  parabola  at  a  given  point, 
we  have  the  following 

RULE. — Produce  the  axis,  and  lay  off  on  it  from  the  focus  a  dis 
tance  equal  to  the  distance  from  the  focus  to  the  point  of  contact. 
The  line  drawn  through  the  point  thus  determined  and  the  given 
point  will  be  the  required  tangent. 


48  CONIC    SECTIONS. 

PROPOSITION    V.— THEOREM. 

The  perpendicular  distance  from  the  focus  of  a  parabola  to 
any  tangent  to  the  curve,  is  a  mean  proportional  between  the 
distance  from  the  focus  to  the  vertex  and  the  distance  from 
the  focus  to  the  point  of  contact. 

In  the  figure  of  tlie  preceding  proposi- 
tion  draw  in  addition  the  vertical  tangent 
Vt;  then  we  are  to  prove  that  Ft2= 


VF-FP.    Because  TtF   and  VFt   are  f  H  y  g  D c 

similar  right-angled  triangles,  we  have 

TF  :Ft::Ft:  VF.    But  TF=PF,  (Prop.  4) ; 
therefore,  PF :  Ft :  :  Ft :  VF 

Whence,  Wf^PF.  VF 

Hence,  the  theorem ;  the  perpendicular  distance  from^etc. 

PROPOSITION    YI.— THEOREM. 

The  sub-tangent  on  the  axis  of  the  parabola  is  bisected  at 
the  vertex. 

In  the  figure  which  is  constructed  as 
in  the  two  preceding  propositions,  draw 
in  addition  the  ordinate  PD,  from  the 
point  of  contact  to  the  axis ;  then  we  T  H  v  F  D 
are  to  prove  that  TD  is  hisected  at  the  vertex  V. 

The  two  right-angled  triangles  TFt  and  tFP  have  the 
side  Ft  common,  and  the  angle  FTt  equal  to  the  angle 
FPt ;  hence  the  remaining  angles  are  equal,  and  the  tri 
angles  themselves  are  equal;  therefore  tT=tP.  From  the 
similar  triangles  TDP,  TVt,  we  have  the  proportion 
Tt:  tP:  :  TV:  VD 

But  tT=tP;  whence  TV=  VD 

Hence  the  theorem ;  the  sub-tangent  on  the  axis,  etc. 


THE    PARABOLA.  49 

Cor.  Since  TV=±TD,  it  follows  that  Vt^PD.  That 
is,  The  part  of  the  vertical  tangent  included  between  the  vertex 
and  any  tangent  line  to  the  parabola,  is  equal  to  one  half  of  the 
ordinate  to  the  axis  from  the  point  of  contact 

PROPOSITION    VII.— THEOREM. 

The  sub-normal  is  equal  to  twice  the  distance  from  the  focus 
to  the  vertex  of  the  parabola* 

In  the  figure  (which  is  the  same  as  that    B 
of  the  last  three  propositions),  PC  is  a 
normal  to  the  parabola  at  the  point  (7,      . 


and  DC  is  the  sub-normal ;  it  is  to  be  T  H  v  F  D 
proved  that  DC=2FV. 

Because  BH  and  PD  are  parallel  lines  included  be 
tween  the  parallel  lines  BP  and  HD,  they  are  equal. 
BF  and  PC  are  also  parallel,  since  each  is  perpendicular 
to  the  tangent  PT ;  hence  BF=PC,  and  also  the  two  tri 
angles  HBF  and  DPC  are  equal. 

Therefore  HF=DC; 

but  HF=2FV; 

whence  DC=2FV. 

Hence  the  theorem ;  the  sub-normal  is  equal  to  twice,  etc. 

SCHOLIUM. — This  proposition  suggests  another  easy  process  for 
constructing  a  tangent  to  a  parabola  at  a  given  point. 

RULE. — Draw  an  ordinate  to  the  axis  from  a  given  point,  and 
from  the  foot  of  this  ordinate  lay  off  on  the  axis,  in  the  opposite 
direction  of  the  vertex,  twice  the  distance  from  the  focus  to  the 
vertex.  Through  the  point  thus  determined  and  the  given  point 
draw  a  line,  and  it  will  be  the  required  tangent. 

PROPOSITION   YII  I.— THEOREM. 

Any  ordinate  to  the  axis  of  a  parabola  is  a  mean  proportion 
al  between  the  corresponding  sub-tangent  and  sub-normal. 
5  D 


50  CONIC    SECTIONS 

Assume  any  point,  as  P,  in  the  parabo 
la  of  which.  F  is  the  focus  and  HB  the 
directrix.  Through  this  point  draw  the 
tangent  PT,  the  normal  P(7,  and  the  or-  T  H  v  F  D  c" 
dinate  PD  to  the  axis.  Then  in  reference  to  the  point  P, 
TD  is  the  sub-tangent,  and  1)  0  the  sub-normal  on  the 
axis ;  and  we  are  to  prove  that 

TD  :  PD  :  :  PD  :  DC 

The  triangle  TFC  is  right-angled  at  P,  and  PD  is  a 
perpendicular  let  fall  from  the  vertex  of  this  angle  upon 
the  hypotenuse.  Therefore,  PD  is  a  mean  proportional 
between  the  segments  of  the  hypotenuse,  (Th.  25,  B.  II, 
Geom.) 

Hence  the  theorem ;  any  ordinate  to  the  axis,  etc. 

SCHOLIUM  1. — For  a  given  parabola,  the  fourth  term  of  the  pro 
portion,  TD  :  PD  :  :  PD  :  DC,  is  a  constant  quantity,  and  equal 
to  twice  the  distance  from  the  focus  to  the  vertex,  (Prop.  7).  By 
placing  the  product  of  the  means  of  this  proportion  equal  to  the 
product  of  the  extremes,  we  have 

PZ>2~  TD-DC=±TD-2DC,  which  may  be  again  resolved  into  the 
proportion 

\TD\PV\  :PD:2DO 

Or,  VD:PD  :  :PD:2DG 

But  VD  is  the  abscissa,  and  PD  is  the  ordinate  of  the  point  P ; 
hence  (Def.  8)  2DC  is  the  parameter  of  the  parabola,  and  is  equal 
to  four  times  the  distance  from  the  focus  to  the  vertex,  or  to  twice 
the  distance  from  the  focus  to  the  directrix. 

SCHOLIUM  2. — If  we  designate  the  ordinate  PD  by  y,  the  abscissa 
VD  by  X,  and  the  parameter  by  2p,  the  above  proportion  becomes 
x^ :  y  :  :  y  :  2p 

Whence,  y  =2px. 

This  equation  expresses  the  general  relation  between  the  abscissa 
and  ordinate  of  any  point  of  the  curve,  and  is  called,  in  Analytical 
Geometry,  the  equation  of  the"  parabola  referred  to  its  principal  ver 
tex  as  an  origin. 

Cor.  The  sub-normal  in  the  parabola  is  equal  to  one-half  of  the 
parameter. 


p 


THE   PARABOLA.  51 

PROPOSITION   IX.—  THEOREM. 

The  parameter,  or  latus  rectum,  of  the  parabola  is  equal  to 
twice  that  ordinate  to  the  axis  which  passes  through  the  focus. 

Let  F  be  the  focus,  and  BB'  the  direc 
trix  of  a  parabola  ;   and  through  the  focus  B 
draw  a  perpendicular  to  the  axis  intersecting  R 
the  curve  at  P  and  Pr.  From  P  and  P'  let  fall 
the  perpendiculars  P_B,  P'B'  ,  on  the  direc- 
trix.     Then  will  ZPF  be  equal  to  2FH,  or 
to  the  parameter  of  the  parabola. 

By  the  definition  of  the  parabola,  PF—PB;  and  be 
cause  PP'  and  BB'  are  parallel,  and  the  parallels  PB  and 
FH  are  included  between  them,  we  have  PB=FH. 

Hence  PF=FH,  or  2PF=2FH=  the  parameter.  Scho.  1, 
Prob.  8. 

Cor.  Since  the  axis  bisects  those  chords  of  the  parabola 
which  are  perpendicular  to  it,  FP=FP].     That  is, 
FP  ;  therefore  PP'=2FH.     That  is, 

The  parameter  of  the  parabola  is  equal  to  the  double  ordi 
nate  through  the  focus. 

PROPOSITION  X.—  THEOREM. 

The  squares  of  any  two  ordinates  to  the  axis  of  a  parabola 
are  to  each  other  as  their  corresponding  abscissas. 

Let  y  and  yr  denote  the  ordinates,  and  x  and  xf  the 
abscissas  of  any  two  points  of  the  parabola;  then,  by 
Scho.  2,  Prop.  8,  we  have  the  two  following  equations  : 

y2=2px  and  y'2=2px' 

Dividing  the  first  of  these  equations  by  the  second, 
member  by  member,  we  have 


Whence  y*  :  y'2  :  :  x  :  xr 

Hence  the  theorem  ;  the  squares  of  any  two  ordinates,  etc. 


52  CONIC    SECTIONS. 

PEOPOSITION   XI.— TIIEOREM. 

If  a  perpendicular  be  drawn  from  the  focus  of  a  parabola 
to  any  tangent  line  to  the  curve,  the  intersection  of  the  perpen 
dicular  with  the  tangent  will  be  on  the  vertical  tangent. 

Let  F  be  the  focus,  and  BH  the  di-  B| 
rectrix  of  the  parabola,  and  PT  a  tan 
gent  to  the  curve  at  the  point  P.     From 
jFdraw  FB  perpendicular  to  the  tangent,  T  H  v  r  D       c~ 
intersecting  it  at  t,  and  the  directrix  at  J3.     We  will  now 
prove  that  the  point  t  is  also  the  intersection  of  the  ver 
tical  tangent  with  the  tangent  PT. 

Because  the  triangle  TFP  is  isosceles,  the  perpendicu 
lar  Ft  bisects  the  base  PT;  therefore  tP=tT.  Again, 
since  Vt  and  DP  are  both  perpendicular  to  the  axis,  they 
are  parallel,  and  the  vertical  tangent  divides  the  sides  of 
the  triangle  TDP  proportionally. 

Hence,  TV:  VD::  Tt :  tP;  but  TV=  YD  (Prop.  6) 
therefore,  Tt=tP. 

That  is,  the  tangent  PT  is  bisected  by  both  the  perpen 
dicular  let  fall  upon  it  from  the  focus,  and  the  vertical 
tangent.  Therefore  the  tangent  PT,  the  vertical  tangent 
and  the  perpendicular  FB,  meet  in  the  common  point  t. 

Hence  the  theorem ;  if  a  perpendicular  be  drawn,  etc. 

PROPOSITION  XI  I.    THEOREM. 

The  parameter  of  the  paraboja  is  to  the  sum  of  any  two  or- 
dinates  to  the  axis,  as  the  difference  of  those  ordlnates  is  to  the 
difference  of  the  corresponding  abscissas. 

Take  any  two  points,  as  P  and  Q,  in  the  parabola  repre 
sented  in  the  following  figure,  and  through  these  points 
draw  the  double  or  dinates  Pp  and  Qq.  VD  and  VE  are 
the  corresponding  abscissas. 

Draw  PS  and  pt  parallel  to  the    axis.      Then,  since 


THE    PARABOLA.  53 

PD=Dp  and  QE=Eq,  we  have  QE+PD  C^ 

=  Qt,  equal  to  the  sum  of  the  two  ordinates ; 
and  QE—PD=  QS,  equal  to  their  differ 
ence;  also   VE—VD=DE,  equal  to  the  v( 
difference  of  the  corresponding  abscissas. 
We  are  now  to  prove  that  pv 

2p  :  Qt :  :  QS  :  DE  *^^ 

in  which  2p  denotes  the  parameter  of  the  parabola. 

Because  PD  and  QE  are  ordinates  to  the  axis,  we  have 
(Scho.  2,  Prop.  8) 

PD*=2p'  VD  (1) 


and  QE=2p-VE  (2) 

Whence         QE*— Pff==2p  (VE—  VD)=2p-DE    (3) 
But         QE2—PTf=  ( QE+  PD}  ( QE—PD)=  Qt-  QS, 

therefore         Qt-QS=2p-J)E  (4) 

Whence  2p  :  Qt :  :  QS  :  DE 

Hence  the  theorem ;  the  parameter  of  the  parabola,  etc. 

Cor.  By  dividing  eq.  (4)  by  eq.  (2),  member  by  member, 
we  obtain 

Qt-QS_DE 


Whence  VE :  DE :  :  QE* :  Qt-  QS 


PROPOSITION   XIII.— THEOREM. 

If  a  tangent  line  be  drawn  to  a  parabola  at  any  point,  and  from 
any  point  of  the  tangent  a  line  be  dravm  parallel  to  the  axis 
terminating  in  the  double  ordinate  from  the  point  of  contact, 
this  line  will  be  cut  by  the  curve  into  parts  having  to  each  other 
the  same  ratio  as  the  segments  into  which  it  divides  the  double 
ordinate. 


54  CONIC    SECTIONS. 

Take  any  point  as  P  in  the  parabo 
la  represented  in  the  figure,  and  of 
which    VD  is  the  axis,  and  through 
this  point  draw  the  tangent  PTto  the 
curve,  and  the  double  ordinate  PQ  to 
the  axis.     Assume  a  point  in  the  tan 
gent  at  pleasure,  as  JL,  and  through  it  PJ 
draw  AC  parallel  to  the  axis,  cutting// 
the  curve  at  B  and  the  double  ordinate  at  C.     Then  we 
arc  to  prove  that 

AB:BC::PC:  CQ 
By  similar  triangles  we  have 

PC:  CA  ::  PD:  DT;  but  DT=2DV(Prop.  6) 
therefore         PC:  CA  :  :  PD  :  2DV  (1) 

But  D  V:  PD  :  :  PD  :  2p  (Scho.  2,  Prop.  8) 

or  2DF:  PD  :  :  2PD  :  2p. 

Inverting  terms,  PD  :  2DV:  :  2p  :  2PD=PQ     (2) 
By  comparing  proportions  (1)  and  (2),  we  get 

PC:  CA::2p:  PQ 

But  2p  :  CQ::  PC:  BC        (Prop.  12) 

Multiplying  the  last  two  proportions,  term  by  term,  we 
have 

2p-PC:  CA-CQ  :  :  2p-PC:  BC'PQ 
The  first  and  third  terms  of  this  proportion  are  equal ; 
therefore  the  second  and  fourth  are  also  equal.     Hence 
we  have  the  proportion 

CA:  BC::  PQ  :  CQ 

Whence  by  division,     CA—BC  :BC::  PQ—CQ  :  CQ 
or  AB:BC:  :  PC:  CQ 

If  we  take  any  other  point,  H,  on  the  tangent,  and 
through  it  draw  the  line  HL  parallel  to  the  axis,  inter 
secting  the  curve  at  K  and  the  ordinate  at  L9  we  will 
have,  in  like  manner, 

HK:  KL:  :  PL:  LQ 
Hence  the  theorem ;  if  a  tangent  be  drawn,  etc. 


THE    PARABOLA. 


55 


PROPOSITION   XIV .— T  H  E  O  R  E  M  . 

If  any  two  points  be  taken  on  a  tangent  line  to  a  parabola,  and 
through  these  points  lines  parallel  to  the  axis  be  drawn  to  meet 
the  curve,  such  lines  will  be  to  each  other  as  the  squares  of  the 
distances  of  the  points  from  the  point  of  contact. 

The  figure  and  construction  being 
the  same  as  in  the  foregoing  proposi 
tion,  we  are  to  prove  that 

AB  :  HK :  :  'PA2  :  PH2 
We  have 
AB :  BC  :  :  PC  :  CQ  (1)  (Prop.  13.) 

Multiplying  the  terms  of  the  second  PJ 
couplet  of  this  proportion  by  PC,  it/ 
becomes 

AB:  BC::~PC2 :  POCQ_       (2) 

But,  (Cor.  Prop.  12)     VD:BC::  ~P1?  :  PC-  CQ    (3) 

Dividing  proportion  (2)  by  proportion  (3),  term  by  term, 
we  have 

AB    ,..l£2.i 

YD  '       '  PI?  ' 

Whence,  AB :VZ>::  ~PC* :  PI?      (4) 

From  the  similar  triangles,  APC  and  TPD,  we  get  the 
proportion 

*  •**  rt  . A  n  . o  St?\ 

PA*:PT  ::PC  :PD 

By  comparing  proportions  W  and  (5)  we  find 

AB  :  YD  :  :  ~FA 
In  like  manner  we  can  prove  that 
HK  :  VD::  PH2 : 
Dividing  proportion  (6)  by  proportion  (7),  term  by  term, 
we  have 

^:l::S:l 

MR'       'PH*' 

Whence,         AB  :  HK :  :  PA' :  Pjf 

Hence  the  theorem ;  if  any  two  points  be  taken,  etc. 


56 


CONIC    SECTIONS. 


APPLICATION. — Conceive  PH  to  be  the  direction  in  which  a  body 
thrown  from  the  surface  of  the  earth,  would  move,  if  it  were  undis 
turbed  by  the  resistance  of  the  air  and  by  the  force  of  gravity.  It 
would  then  move  along  the  line  PH,  passing  over  equal  spaces  in 
equal  times.  When  a  body  falls  under  the  action  of  gravity,  one  of 
the  laws  of  its  motion  .is,  that  the  spaces  are  proportional  to  the  squares 
of  the  times  of  descent ;  hence,  if  we  suppose  gravity  to  act  upon 
the  body  in  the  direction  AC,  the  lines  AJ3,  TV,  HK,  etc.,  must 
be  to  each  other  as  the  squares  PA  ,  PT  ,  PH  ,  etc. ;  that  is,  the 
real  path  of  a  projectile  in  vacuo,  possesses  the  property  of  the 
parabola  that  has  been  demonstrated  in  this  proposition.  In  other 
words, 

The  path  of  a  projectile,  undisturbed  by  the  resistance  of  the  airy 
is  a  parabola,  more  or  less  curved,  depending  upon  the  direction  and 
intensity  of  the  projectile  force. 


PEOPOSITION  XY.— THEOBEM. 

The  abscissas  of  any  diameter  of  the  parabola  are  to  each 
other  as  the  squares  of  their  corresponding  ordinates. . 

Let  P  be  any  point  on  a  parabola, 
PL  a  tangent  line,  and  PF  a  diame 
ter  through  this  point.  From  the 
points  B,  V,K,  etc.,  assumed  at  pleas 
ure  on  the  curve,  draw  ordinates  and 
parallels  to  the  diameter,  forming  the 
quadrilaterals  PCBA,  PD  VT,  etc. 

!N"ow,  since  the  ordinates  to  any  di 
ameter  of  the  parabola  are  parallel  to 
the  tangent  line  through  the  vertex  of  that  diameter, 
these  quadrilaterals  are  parallelograms  and  their  opposite 
sides  are  equal.  But,  by  the  preceding  proposition,  we 
have 

AB  :  TV:  HK,  etc.,  :  :~PA2 :  ~PT2 :  PH\  etc. 
or        PCiPD:  PE,  etc.,  : : ~£C2  : ~Vff  :  KE*>  etc. 


THE    PARABOLA. 


57 


By  definition  6,  PC  is  the  ordlnate  and  BC  the  abscis 
sa  of  the  point  B,  and  so  on. 
Hence  the  theorem  ;  the  abscissas  of  any  diameter,  etc. 


PROPOSITION     XY  I.— THEOREM. 

If  a  secant  line  be  drawn -parallel  to  any  tangent  line  to  the 
\parabola,  and  ordinates  to  the  axis  be  drawn  from  the  point  of 
contact  and  the  two  intersections  of  the  secant  with  the  curve, 
these  three  ordinates  will  be  in  arithmetical  progression. 

Let  CT  be  the  tangent  line  to  the 
parabola,  and  EH  the  parallel  secant. 
Draw  the  ordinates  EG,  CD,  and 
HI,  to  the  axis  VI,  and  through  E 
draw  EK  parallel  to  VI. 

We  are  now  to  prove  that 


The  similar  triangles,  HKE'and.  CDT,  give  the  pro- 
p jrti  n 

HK  :  KE::  CD:  DT=2VD 
and,  by  proposition  12,  we  have 

2p:KL:  :  HK :  KE. 
Therefore         2p  :  KL  :  :  CD  :  2  VD,  (1) 

and  from  the  equation,  y*=2px,  we  get,  by  making  y=  CD 
and  x=  VD, 

2p:  2OD:  :  CD:  2FZ>  (2) 

By  dividing  proportion  (1)  by  (2),    term  by  term,  we 
shall  have 

KL 


"Whence  KL=2CD 

But  KL=HI+KI=HI+EG; 

therefore  HI+  EG=  2  CD 

Hence  the  theorem;  if  a  secant  line  be  drawn,  etc. 


58  CONIC    SECTIONS. 

SCHOLIUM  1.  —  If  we  draw  CM  parallel,  and  MN  perpendicular 
to  VI,  then  2CD=2MN=EG-}-HI;  and  since  MNis  parallel  to 
each  of  the  lines  EG  and  HI,  the  point  M  bisects  the  line  EH. 
That  is,  the  diameter  through  G  bisects  its  ordinate  EH)  and  as 
HE  is  any  ordinate  to  this  diameter,  it  follows  that 

A  diameter  of  the  parabola  divides  into  equal  parts  all  chords  of 
the  curve  parallel  to  the  tangent  through  the  vertex  of  the  diameter. 

SCHOLIUM  2.  —  Hence,  as  the  abscissas  of  any  diameter  of  the 
parabola  and  their  ordinates  have  the  same  relations  as  those  of  the 
axis,  namely;  that  the  ordinates  are  bisected  by  the  diameter,  and 
their  squares  are  proportional  to  the  abscissas  ;  so  all  the  other  prop 
erties  of  this  curve,  before  demonstrated  in  reference  to  the  abscis 
sas  and  ordinates  of  the  axis,  will  likewise  hold  good  in  reference  to 
the  abscissas  and  ordinates  of  any  diameter. 

PROPOSITION  XVII  .—  T  H  E  0  E  E  M  . 

The  square  of  an  ordinate  to  any  diameter  of  the  parabola 
is  equal  to  four  times  the  product  of  the  corresponding  abscissa 
and  the  distance  from  the  vertex  of  that  diameter  to  the  focus. 

Let  "FJTbe  th.e  axis  of  aparaola, 
and  through  any  point,  as  P,  of  the 
curve,  draw  the  tangent  P  T,  and 
the  diameter  PW;  also  draw  the 
secant  Qq,  parallel  PT,  and  pro 
duce  the  ordinate  QN,  and  the  di 
ameter  P  W,  to  meet  at  D.  From  the  focus  let  fall  the 
perpendicular  FY  upon  the  tangent,  and  draw  FP  and 
VY.  We  are  now  to  prove  that 


Because  FYis  perpendicular  to  PT,  Qv  parallel  to  PT 
and  DQ  parallel  to  each  of  the  lines  PM  and  VY,  the 
triangles  DQv,  PMT,  TFFand  TFFare  all  similar. 


Whence     Qv  :  QI>  :  :  TF  :  TP  (1) 

But       ~TF*=PF*  and  TF=  PF-  VF.     (Prop.  5) 


THE    PARABOLA.  59 

Substituting  these  values  in  proportion  (1)  and  dividing 
the  third  and  fourth  terms  of  the  result  by  PF,  it  becomes 

~~Qvi~Qtf  :  :PF:  VF  (2) 

Again,  from  the  triangles  QDv  and  PM  T  we  get 
QD  :Dv  ::PM:  MT=2VM 

:  :  PM2  :  2PM-  VM 

But  (Scho.  2,  Prop.  8)      PM*=±VF'  VM 
Whence         QD  :  Dv   :  :  4  VF-  VM  :  2PM-  VM; 

:  :  4  F.F  :  2PM 
therefore  2PM  •  §£>=4  VF-Dv  (3) 

By  subtracting  the  equation  QNZ=4t  VF-  F^Vfrom  the 
equation  PM  2=4  VF-  VM,  member  from  member,  we 
have 

4:  VF-  (  VM—  VN) 


Whence  ^ 

(PM+QN]  (PM—QN)=(PM+QN)  DQ=±VF-DP  (4) 

Subtracting  eq.  (4)  from  eq.  (3),  member  from  member, 
we  obtain 

(PM—QN]  i>§=4  VF  (Dv—DP)=±  VF-Pv 
and  because  PM—QN—DQ,  this  last  equation  becomes 


Substituting  this  value  of  JDQ2  in  proportion  (2),  we  have 
§?  :  4VF-  Pv  :  :  PF  :  VF 

or  ^v2  :  4Pv:  :  PF  :  1 

Whence       ~Qv=±PF  -  Pv 

Hence  the  theorem  ;  the  square  of  an  ordinate,  etc. 

Cor.  If,  in  the  course  of  this  demonstration,  we  had 
used  the  triangle  vdq  in  the  place  of  vDQ,  to  which  it  is 
similar,  we  would  have  found  that  qv2=4PF'Pv^  whence 
Qv=qv.  And  since  the  same  may  be  proved  for  any  ordi- 
nate,  it  follows  that 


60  CONIC    SECTIONS. 

All  the  ordinates  of  the  parabola  to  any  of  its  diameters  are 
bisected  by  that  diameter. 

SCHOLIUM. — The  parameter  of  any  diameter  of  the  parabola  has 
been  defined  (Def.  8)  to  be  one  of  the  extremes  of  a  proportion,  of 
which  any  ordinate  to  the  diameter  is  the  mean  and  the  corresponding 
abscissa  the  other  extreme. 

Now,  we  have  just  shown  that     Qv  =qv  =^PF'Pv. 

Whence,  Pv  :  Qv  :  :  Qv  :  4PF.  4PF,  which  remains  constant 
for  the  same  diameter,  is  therefore  the  parameter  of  the  diameter 
PW.  And  as  the  same  may  be  shown  for  any  other  diameter,  we 
conclude  that 

The  parameter  of  any  diameter  of  the  parabola  is  equal  to  four 
times  the  distance  from  the  vertex  of  that  diameter  to  the  focus. 

PKOPOSITION    XYIII.— THEOREM. 

The  parameter  of  any  diameter  of  the  parabola  is  equal  to 
the  double  ordinate  to  this  diameter  that  passes  through  the  focus. 

Through  any  point,  as  P,  of  the  pa-  / Jfy 

rabola  of  which  F  is  the  focus  and  V 
the  vertex,  draw  the  diameter  PW,  the 
tangent  P  T,  and,  through  the  focus  the 
double  ordinate  BD,  to  the  diameter. 
It  is  now  to  be  proved  that  4PF,  or  the 
parameter  to  this  diameter,  is  equal  to  BD. 

Because  PW  is  parallel  to  TX,  and  BD  to  TP,  TPvF 
is  a  parallelogram,  and  Pv—  TF.  But  PF=FT  (Prop.  4), 
hence  Pv=PF. 

By  the  preceding  proposition,  Bv=4tPF-Pv  =4PF-PF 

Whence,  Bv=2PF ;  therefore,  2Bv=J3D=£PF 

i 

Hence  the  theorem ;  the  parameter  of  any  diameter,  etc. 

PROPOSITION    XIX.-THEOREM. 

The  area  of  the  portion  of  the  parabola  included  between 
the  curve,  the  ordinate  from  any  of  its  points  to  the  axis,  and 


THE    PARABOLA.  61 

the  corresponding  abscissa,  is  equivalent  to  two  thirds  of  the 
rectangle  contained  by  the  abscissa  and  ordinate. 

Let  VD  be  the  axis  of  a  parabo 
la,  and  VIP  any  portion  of  the 
curve.  Draw  the  extreme  ordinate 
PI)  to  the  axis,  and  complete  the 
rectangle  VAPD ;  then  will  the 
area  included  between  the  curve 
VIP,  the  ordinate  PD,  and  the  abscissa  FD,  be  equiva 
lent  to  two  thirds  of  the  rectangle  VAPD. 

Take  any  point  J,  between  P  and  the  vertex,  and  draw 
PI,  producing  it  to  meet  the  axis  produced  at  E. 

Now,  from  the  similar  triangles,  PQI  and  PDEy  we 
get  the  proportion 

PQ:  QI:  :  PD  :  DE: 

Whence      PQ  -  DE=  QI  •  PD=  GD  -  PD.          (l) 

If  we  suppose  the  point  I  to  approach  P,  the  secant  line 
PJE  will,  at  the  same  time,  approach  the  tangent  PT;  and 
finally,  when  I  comes  indefinitely  near  to  P,  the  secant 
will  sensibly  coincide  with  the  tangent  PT,  and  DE  may 
then  be  replaced  by  DT=2DV=2PA.  Under  this  sup 
position,  eq.  (1)  becomes 

2PQ  -  PA=PD  -  GD. 

That  is,  when  the  rectangles  GDPH  and.  APQ  C  become 
indefinitely  small,  we  shall  have 

Eect.  GDPH=2HQct.  APQC. 

We  will  call  Kect.  GDPH  the  interior  rectangle,  and 
Eect.  APQC  the  exterior  rectangle.  If  another  point  be 
taken  very  near  to  J,  and  between  it  and  the  vertex,  and 
with  reference  to  it  the  interior  and  exterior  rectangles  be 
constructed  as  before,  we  should  again  have  the  interior 
equivalent  to  twice  the  exterior  rectangle.  L  et  us  conceive 
this  process  to  be  continued  until  all  possible  interior  and 
exterior  rectangles  are  constructed  ;  then  would  we  have 
Sum  interior  rectangles=2  sum  exterior  rectangles. 


62  CONIC    SECTIONS. 

But,  under  the  supposition  that  these  rectangles  are  in 
definitely  small,  the  sum  of  the  interior  rectangles  be 
comes  the  interior  curvilinear  area,  and  the  sum  of  the 
exterior  rectangles  the  exterior  curvilinear  area,  and  the 
two  sums  make  up  the  rectangle  APD  F.  Therefore,  if 
this  rectangle  were  divided  into  three  equal  parts,  the  in 
terior  area  would  contain  two  of  these  parts. 

Hence  the  theorem ;  the  area  of  the  portion  of  the,  etc. 

PROPOSITION  XX.— THEOREM. 

If  a  parabola  be  revolved  on  its  axis,  the  solid  generated 
will  be  equivalent  to  one  half  of  its  circumscribing  cylinder. 

Conceive  the  parabola  in  the  fig- 
ure,  which  is  constructed  as  in  the 
last  proposition,  to  revolve  on  its 
axis  VD.  We  are  then  to  find  the 
measure  of  the  volume  generated.  T  E  V^G ~i> 

The  rectangle  ID  will  generate 
a  cylinder  having  D  Q  for  the  radi 
us  of  its  base,  and  DGr  for  its  axis;  and  the  rectangle  AI 
will  generate  a  cylindical  band,  whose  length  is  CT,  and 
thickness  PQ. 

The  solidity  of  the  cylinder  =nDQ*^D& 

The  solidity  of  the  band  =7r(PZ)2— ~DQ*)  •  F£= 
x[PD*—(PD—PQy]  •  7<3W[2PD  -  PQ— P§2]  •  VG 

E~ow,  under  the  supposition  that  the  point  1  is  indefi 
nitely  near  to  P,  DQ  may  be  replaced  by  PD,  VG  by  FD,  - 
and  PQ2  may  be  neglected  as  insensible  in  comparison 
with  2PD-PQ.  These  conditions  being  introduced  in 
the  above  expressions  for  the  solidities  of  the  cylinder  and 
band,  they  become 

The  solidity  of  the  cylinder =7rPZ)2  •  DG 

The  solidity  of  the  band        =  2nPD  -PQ-VD 


THE    PARABOLA.  63 

• 

"Whence, 

sol.  of  cylinder  :  sol.  of  band  :  :  ~TI?  -DG:  2PD-  PQ'  VD  (1) 
But,  when  /and  P  are  sensibly  the  same  point, 

PQ  :  GD  :  :  PD  :  2  VD 
therefore, 


The  terms  in  the  last  couplet  of  proportion  (1)  are  there 
fore  equal,  and  we  have 

sol.  of  cylinder  :  sol.  of  band  :  :  1  :  1 

or        sol.  of  cylinder=sol.  of  band. 

In  the  same  manner  we  may  prove  that  any  other  inte 
rior  cylinder  is  equivalent  to  the  corresponding  exterior 
band.  Hence  the  sum  of  all  the  possible  interior  solids 
is  equivalent  to  the  sum  of  the  exterior  solids.  But  the 
two  sums  make  up  the  cylinder  generated  by  the  rectan 
gle  VDPA;  therefore  either  sum  is  equivalent  to  one 
half  of  the  cylinder. 

Hence  the  theorem  ;  if  a  parabola  be  revolved,  etc. 

REMARK.  —  The  body  generated  by  the  revolution  of  a  parabola 
about  its  axis  is  called  a  Paraboloid  of  Revolution. 

PROPOSITION   XXI.—  THEOREM. 

If  a  cone  be  cut  by  a  plane  parallel  to  one  of  its  elements, 
the  section  will  be  a  parabola. 

Let  M  VN  be  a  section  of  a  cone  by  a 
plane  passing  through  its  axis,  and  in  this 
section  draw  AH  parallel  to  the  element  VM.  K 
Through  AH  conceive  a  plane  to  be  passed 


perpendicular  to  the  plane  M  VN;  then  will  M"]t-^|-- 
the  section  DA  GI  of  the  cone  made  by  this  last  plane, 
be  a  parabola.  In  the  plane  MVN,  draw  MN  and 
KL  perpendicular  to  the  axis  of  the  cone,  and  through 
them,  pass  planes  perpendicular  to  this  axis.  The 
sections  of  the  cone,  by  these  planes,  will  be  circles, 


64  CONIC    SECTIONS. 

of  which  MN  and  KL,  respectively,  are  the  diameters. 
Through  the  points  F  and  H,  in  which  AH  meets  KL 
and  MN,  draw  in  the  section  DA  GI  the  lines  FG  and 
HI,  perpendicular  to  AH.  Because  the  planes  DAI  and 
MVN  are  at  right  angles  to  each  other,  FG  is  perpendic 
ular  to  KL,  and  HI  is  perpendicular  to  MN. 

Now,  from  the  similar  triangles  AFL,  AHN,  we  have 

AF:AH::FL:HN  (1) 

By  reason  of  the  parallels,  KF=MH;  multiplying  the 
first  term  of  the  second  couplet  of  proportion  (1)  by  KF, 
and  the  second  term  by  MH,  it  becomes 

AF:  AH:  :  FL-KF:  HN'MH  (2) 

But  FG-  is  an  or  din  ate  of  the  circle  of  which  KL  is 
the  diameter,  and  HI  an  ordinate  of  the  circle  of  which 
JOT  is  the  diameter:  therefore 

FL-KF=FG\  and  JZ2V-JfJJ=S?  (Cor.,  Th.  IT,  B.  Ill, 

Geom.) 

Substituting,  for  the  terms  of  the  second  couplet,  in  pro 
portion  (2),  these  values,  it  becomes 

A  F  :  AH  :  :  FG2  :  ~H12 


This  proportion  expresses  the  property  that  was  dem 
onstrated  in  proposition  15  to  belong  to  the  parabola. 
Hence  the  theorem  ;  if  a  cone  be  cut  by  a  plane,  etc. 

Cor.  From  the  proportion,  AF:  AH:  :  FG2  :  HI2  we 


,  . 

get       j-1=  -TTT',  that  is,    -r-=p  or   -jjj-  which  is  a  third 

proportional  to  any  abscissa  and  the  corresponding  ordi 
nate  of  the  section,  is  constant,  and  (by  Def.  8)  is  the  para 
meter  of  the  section. 


THE   HYPERBOLA.  65 


THE  HYPERBOLA. 


DEFINITIONS. 

1.  The  Hyperbola  is  a  plane   curve,  generated  by  the 
motion  of  a  point   subjected  to  the  condition  that  the 
difference  of  its  distances  from  two  fixed  points  shall  be 
constantly  equal  to  a  given  line. 

REMARK  1. — The  distance  between  the  foci  is  also  supposed  to 
be  known,  and  the  given  line  must  be  less  than  the  distance  between 
the  fixed  points }  that  is,  less  than  the  distance  between  the  foci. 

REMARK  2. — The  ellipse  is  a  curve  confined  by  two  fixed  points 
called  the  foci ;  and  the  sum  of  two  lines  drawn  from  any  point  in 
the  curve  is  constantly  equal  to  a  given  line.  In  the  hyperbola,  the 
difference  of  two  lines  drawn  from  any  point  in  the  curve,  to  the 
fixed  points,  is  equal  to  the  given  line.  The  ellipse  is  but  a  single 
curve,  and  the  foci  are  within  it ;  but  it  will  be  shown  in  the  course 
of  our  investigation,  that 

The  hyperbola  consists  of  two  equal  and  opposite  branches,  and 
the  least  distance  between  them  is  the  given  line. 

2.  The  Center  of  the  hyperbola  is  the  middle  point  of 
the  straight  line  joining  the  foci. 

3.  The  Eccentricity  of  the  hyperbola  is  the  distance  from 
the  center  to  either  focus. 

4.  A  Diameter  of  the  hyperbola  is  a  straight  line  pass 
ing  through  the  center,  and  terminating  in  the  opposite 
branches  of  the  curve.     The  extremities  of  a  diameter 
called  its  vertices. 

6*  E 


66  CONIC    SECTIONS. 

5.  The  Major,  or  Transverse  Axis,  of  the  hyperbola  is 
the  diameter  that,  produced,  passes  through  the  foci. 

6.  The  Minor,  or  Conjugate  Axis,  of  the  hyperbola  bisects 
the  major  axis  at  right-angles;   and  its  half  is  a  mean 
proportional  between  the  distances  from  either  focus  to 
the  vertices  of  the  major  axis. 

7.  An  Ordinate  to  a   diameter  of  the  hyperbola  is  a 
straight  line,  drawn  from  any  point  of  the  curve  to  meet 
the  diameter  produced,  and  is  parallel  to  the  tangent  at 
the  vertex  of  the  diameter. 

8.  An  Abscissa  is  the  part  of  the  diameter  produced  that 
is  included  between  its  vertex  and  the  ordinate. 

9.  Conjugate  Hyperbolas  are  two  hyperbolas  so  related 
that  the  major  and  minor  axes  of  the  one  are,  respectively, 
the  minor  and  major  axes  of  the  other. 

10.  Two  diameters  of  the  hyperbola  are  conjugate,  when 
either  is  parallel  to  the  tangent  lines  drawn  through  the 
vertices  of  the  other. 

The  conjugate  to  a  diameter  of  one  hyperbola  will  ter 
minate  in  the  branches  of  the  conjugate  hyperbola. 

11.  The  Parameter  of  any  diameter  of  the  hyperbola  is 
a  third  proportional  to  that  diameter  and  its  conjugate. 

12.  The  parameter  of  the  major  axis  of  the  hyperbola 
is  called  the  principal  parameter,  the  latus-rectum,  or  simply 
the  parameter  ;  and  it  will  be  proved  to  be  equal  to  the 
chord  of  the  hyperbola  through  the  focus  and  at  right- 
angles  to  the  major  axis. 

EXPLANATORY  REMARKS. — Thus,  let  FT  be 
two  fixed  points.  Draw  a  line  between  them,  and 
bisect  it  in  C.  Take  GAj  CA',  each  equal  to  one 
half  the  given  line,  and  CA  may  be  any  distance 
less  than  CF;  A' A  is  the  given  line,  and  is  called 
the  major  axis  of  the  hyperbola.  Now,  let  us  suppose  the  curve 
already  found  and  represented  by  ADP.  Take  any  point,  as  P,  and 
join  P,  F  and  P}  Fr ;  then;  by  Def.  1,  the  difference  between  PF' 


THE    HYPEEBOLA. 


67 


and  PF  must  be  equal  to  the  given  line  A' A  ;  and  conversely,  if 
PF'— PF=A'A,  then  P  is  a  point  in  the  curve. 

By  taking  any  point,  P,  in  the  curve,  and  joining  P,  F  and  P}  F' 
a  triangle  PFF'  is  always  formed,  having  F'F  for  its  base,  and 
A' A  for  the  difference  of  the  sides ;  and  these  are  all  the  conditions 
necessary  to  define  the  curve. 

As  a  triangle  can  be  formed  directly  opposite  PF' F,  which  shall 
be  in  all  respects  exactly  equal  to  it,  the  two  triangles  having  F'F 
for  a  common  side ;  the  difference  of  the  other  two  sides  of  this 
opposite  triangle  will  be  equal  to  A' A,  and  correspond  with  the  con 
dition  of  the  curve. 

Hence,  a  curve  can  be  formed  about  the  focus  Ff}  exactly  similar 
and  equal  to  the  curve  about  the  focus  F. 

We  perceive,  then,  that  the  hyperbola 
is  composed  of  two  equal  curves  called 
branches,  the  one  on  the  right  of  the  cen 
ter  and  curving  around  the  right-hand 
focus,  and  the  other  on  the  left  of  the 
center  and  curving  around  the  left-hand 
focus.  In  like  manner,  by  making  CB 
equal  to  a  mean  proportional  between 
FA  and  FA',  and  constructing  above  and  below  the  center  the 
branches  of  the  hyperbola  *of  which  BB'=ZCB  is  the  major,  and 
A  A'  the  minor  axis,  we  have  the  hyperbola  which  is  conjugate  to 
the  first.  PP  is  a  diameter  of  the  hyperbola,  PT  a  tangent  line 
through  the  vertex  of  the  diameter,  and  QQ',  parallel  to  PT  and 
terminating  in  the  branches  of  the  conjugate  hyperbola,  is  conjugate 
to  the  diameter  PP'.  IID  is  the  ordinate  from  the  point  H  to  the 
diameter  CP,  and  PD  is  the  corresponding  abscissa. 


PROPOSITION    I. -PROBLEM. 

To  describe  an  hyperbola  mechanically. 

Take  a  ruler,  F'H,  and  fasten  one  end  at  the  point  JP,  on 
which  the  ruler  may  turn  as  a  hinge.  At  the  other  end,  at 
tach  a  thread,  the  length  of  which  is  less  than  that  of  the 


68  CONIC    SECTIONS. 

ruler  by  the  given  line  A' A.  Fas 
ten  the  other  end  of  the  thread  at  F. 
"With  the  pencil,  P,  press  the  thread 
against  the  ruler,  and  keep  it  at 
equal  tension  between  the  points  H 
and  F.  Let  the  ruler  turn  on  the 
point  F',  keeping  the  pencil  close  /  \ 

to  the  ruler  and  letting  the  thread  slide  round  the  pen 
cil  ;  the  pencil  will  thus  describe  a  curve  on  the  paper. 

If  the  ruler  be  changed,  and  made  to  revolve  about  the 
other  focus  as  a  fixed  point,  the  opposite  branch  of  the 
curve  can  be  described. 

In  all  positions  of  P,  except  when  at  A  or  A ',  PFr  and 
PF  will  be  two  sides  of  a  triangle,  and  the  difference  of 
these  two  sides  is  constantly  equal  to  the  difference  be 
tween  the  ruler  and  the  thread ;  but  that  difference  was 
made  equal  to  the  given  line  A' A  ;  hence,  by  Definition 
1,  the  curve  thus  described  must  be  an  hyperbola. 

Cor.  From  any  point,  as  P,  of  the  hyperbola,  draw  the 
ordinate  PD  to  the  major  axis,  and  produce  this  ordinate 
to  P',  making  DP'  equal  to  PD;  and  draw  FP,  FP', 
F'P  and  F'P'.  Then,  because  F'D  is  a  perpendicular  to 
PP  at  its  middle  point,  we  have  FP=FP',  and  F*P=* 
F'P' ;  whence 

F'P—FP=F'P'—FP',  and  P'  is  a  point  of  the  hyper 
bola.  Therefore,  PP'  is  a  chord  of  the  hyperbola  at  right 
angles  to  the  major  axis,  and  is  bisected  by  this  axis ;  and 
as  the  same  may  be  proved  for  any  other  chord  drawn  at 
right  angles  to  the  major  axis,  we  conclude  that 

All  chords  of  the  hyperbola  which  are  drawn  at  right  angles 
to  the  major  axis  are  bisected  by  that  axis.  It  may  be  proved, 
in  like  manner,  that 

All  chords  of  the  hyperbola  which  are  drawn  at  right  angles 
to  the  conjugate  axis  are  bisected  by  that  axis. 


THE   HYPERBOLA.  69 

PROPOSITION  II.— THEOEEM. 

If  a  point  be  taken  within  either  branch  of  the  hyperbola,  or 
on  the  concave  side  of  the  curve,  the  difference  of  its  distances 
from  the  foci  will  be  greater  than  the  major  axis;  and  if  a 
point  be  taken  without  both  branches,  or  on  the  convex  side  of 
both  curves,  the  difference  of  its  distances  from  the  foci  will  be 
less  than  the  major  axis. 

Let  A  A'  be  the  major  axis,  and 
F  and  Ff  the  foci  of  an  hyperbola. 
Within  the  branch  APX  take  any 
point,  Q,  and  draw  FQ  and  F'Q; 
then  we  are  to  prove  F'A7 A  F 

First.— That  F'Q—FQ  is  greater  than  A  A'. 

Since  Q  is  within  the  branch  APX,  the  line  F'Q  must 
cut  the  curve  at  some  point,  as  P.  Draw  PF  and  FQ. 

By  the  definition  of  the  hyperbola,  FT—  PF=  AA'. 
Adding  PQ+PF  to  both  members  of  this  equation,  it 
becomes 

F'P—PF+PQ+PF=AA'+ PQ+PF 

or,  F'Q=AA'+ PQ+PF. 

But  PQ  and  PF  being  two  sides  of  the  triangle  FPQ, 
are  together  greater  than  the  third  side  FQ.  Therefore 
F'Q>AA'+ FQ;  and,  by  taking  FQ  from  both  members 
of  this  inequality,  we  have 

F'Q-FQ>AA'. 

Second. — Take  any  point,  q,  without  both  branches  of  the 
hyperbola,  and  join  this  point  to  either  focus,  as  F.  Then 
since  q  is  without  the  branch  APF,  the  line  qF  must  cut 
the  curve  at  some  point,  P.  Draw  qF,  qF',  and  PF'. 

Because  P  is  a  point  on  the  curve,  we  have  PFf — PF 
=AA'.  Adding  Pq+PFto  the  members  of  this  equa 
tion  it  becomes 

PFr—PF+  Pq+  PF=  A  A'+ PF+  Pq 

or,          PF'+Pq=AA'+PF+Pq=AA'+qF. 


70  CONIC    SECTIONS. 

But  PFf  and  P/,  being  two  sides  of  the  triangle  F'Pq, 
are  together  greater  than  the  third  side  qF'.  "Whence 
qF'<AAr-\-qF;  and  by  taking  qF  from  both  members 
of  this  inequality,  we  have  qF' — qF<A A'. 

Hence  the  theorem  ;  if  a  point  be  taken,  etc. 

Cor.  Conversely :  If  the  difference  of  the  distances  from 
any  point  to  the  foci  of  an  hyperbola  be  greater  than  the  major 
axis,  the  point  will  be  within  one  of  the  branches  of  the  curve  ; 
and  if  this  difference  be  less  than  the  major  axis,  the  point  will 
be  without  both  branches. 

For,  let  the  point  Q  be  so  taken  that  F'Q— FQ>AAf; 
then  the  point  Q  cannot  be  on  the  curve ;  for  in  that  case 
we  should  have  F'  Q — FQ= A Af.  And  it  cannot  be  with 
out  both  branches  of  the  curve,  for  then  we  should  have 
F'Q — FQ<AA',  from  what  is  proved  above.  But  it  is 
contrary  to  the  hypothesis  that  F'  Q — FQ  is  either  equal 
to  or  less  than  A  A' ;  hence  the  point  Q  must  be  within 
one  of  the  branches  of  the  hyperbola. 

In  like  manner  we  prove  that,  if  the  point  q  be  so  cho 
sen  that  qFf — qF<AA',  this  point  must  be  without  both 
branches  of  the  hyperbola. 


PEOPOSITION  III.— THEOKEM.- 

A  tangent  to  the  hyperbola  bisects  the  angle  contained  by 
lines  drawn  from  the  point  of  contact  to  the  foci. 

Let  F',  F  be  the  fgci,  and  -P 
any  point  on  the  curve;  draw 
PF',  PF  and  bisect  the  angle 
F'PF\>j  the  line  TT' ;  this  line 
will  be  a  tangent  at  P. 

If  TT'  be  a  tangent  at  P,  ev-  F^A7      c     TA"F 
ery  other  point  on  this  line  will  be  without  the  curve. 


THE   HYPERBOLA.  71 

Take  PG=PF  and  draw  GF;  TT'  bisects  GF,  and 
any  point  in  the  line  TT'  is  at  equal  distances  from  F 
and  G  (Scho.  1,  Th.  18,  B.  I,  Geom).  By  the  definition 
of  the  curve,  Ff  Gr=A'A  the  given  line.  ~Now  take  any 
other  point  than  P  in  T T' ,  as  E,  and  draw  EF',  EF  and 
EG. 

Because  JEFia  equal  to  EG  we  have 

EF'—EF=EF'—EG. 

^uiEF'—EG,  is  less  than  F'G,  because  the  differ 
ence  of  any  two  sides  of  a  triangle  is  less  than  the  third 
side.  That  is,  EF' — EF  is  less  than  A' A;  consequent 
ly  the  point  E  is  without  the  curve  (Prop.  2),  and  as  E 
is  any  point  on  the  line  TT ',  except  P,  therefore,  the  line 
TT',  which  bisects  the  angle  at  P,  is  a  tangent  to  the 
curve  at  that  point. 

Hence  the  theorem ;  a  tangent  to  the  hyperbola,  etc. 

SCHOLIUM. — It  should  be  observed  that  by  joining  the  variable 
point,  P,  in  the  curve,  to  the  two  invariable  points,  F'  and  F,  we 
form  a  triangle;  and  that  the  tangent  to  the  curve  at  the  point  P, 
bisects  the  angle  of  that  triangle  at  P. 

But  when  any  angle  of  a  triangle  is  bisected,  the  bisecting  line 
cuts  the  base  into  segments  proportional  to  the  other  sides.  (Th. 
24,  B.  II,  Geom). 

Therefore,  F'P  :  PF=F'T  :  TF 

Kepresent  P'Pby  r'  and  PF  by  r; 

then  r'  :  r=Ff  T  :  TF 

But  as  /  must  be  greater  than  r  by  a  given  quantity,  a, 

therefore,  r+a  :  r=F'T  :  T  F 

Or,  1+^  :  \=F'T  :  TF 

Let  it  be  observed  that  a  is  a  constant  quantity,  and  r  a  variable 
one  which  can  increase  without  limit;  and  when  r  is  immensely  great 

in  respect  to  a,  the  fraction  -  is  extremely  minute,  and  the  first  term 
r 

of  the  above  proportion  would  not  in  any  practical  sense  differ  from 
the  second;  therefore,  in  that  case,  the  third  term  would  not  essen- 


72  CONIC    SECTIONS. 

tially  differ  from  the  fourth;  that  is,  F 'T  does  not  essentially  differ 
from  FT  when  r,  or  the  distance  of  P  from  F'  is  immensely  great. 
Hence,  the  tangent  at  any  point  P,  of  the  hyperbola,  can  never  cross 
the  line  FF'  at  its  middle  point,  but  it  may  approach  within  the  least 
imaginable  distance  to  that  point. 

If,  however,  we  conceive  the  point  P  to  be  removed  to  an  infinite 
distance  on  the  curve,  the  tangent  at  that  point  would  cut  AA'  at 
its  middle  point  C}  and  the  tangent  itself  is  then  called  an  asymptote. 

PKOPOSITION    I  Y.— THEOREM. 

Every  diameter  of  the  hyperbola  is  bisected  at  the  center. 

Let  F  and  F'  be  the  foci,  and 
A  A1  the  major  axis  of  an  hyperbo 
la.  Take  any  point,  as  P,  in  one 
of  the  branches  of  the  curve ;  draw 
PF  and  PF',  and  complete  the 
parallelogram  PFP'F'. 

We  will  now  prove  that  P'  is  a 
point  in  the  opposite  branch  of  the  hyperbola,  and  thai 
PP'  passes  through,  and  is  bisected  at,  the  center,  C. 

Because  PFP'F'  is  a  parallelogram,  the  opposite  sides 
are  equal;  therefore F'P—PF=FP'—P'F';  but  since  J 
is,  by  hypothesis,  a  point  of  the  hyperbola,  F'P — PF— 
AA';  hence  FP— PfF'= AA',  and  P'  is  also  a  point  of 
the  hyperbola. 

Again,  the  diagonals,  F'F,  P'P  of  the  parallelogram, 
mutually  bisect  each  other ;  hence  C  is  the  middle  point 
of  the  line  joining  the  foci,  and  (Def.  2)  is  the  center  of 
the  hyperbola.  P  P'  is  therefore  a  diameter,  and  is  bi 
sected  at  the  center,  C. 

Hence,  the  theorem ;  every  diameter  of  the  hyperbola,  etc. 

PROPOSITION    Y.— THEOREM. 

Tangents  to  the  hyperbola  at  the  vertices  of  a  diameter  are 
parallel  to  each  other.  -  . 


THE    HYPERBOLA. 


73 


At  the  extremities  of  the  diam 
eter,  PP',  of  the  hyperbola  repre 
sented  in  the  figure,  draw  the  tan 
gents  TT'  and  VV.  We  are  now 
to  prove  that  these  tangents  are 
parallel.  By  proposition  (Prop.  3) 
TT  bisects  the  angle  FPF'>  and 
V  V  also  bisects  the  angle  F'P'F.  But  these  angles  being 
the  opposite  angles  of  the  parallelogram  FPF'P',  are 
equal;  therefore  the  [__T'PF=ihe  [__PT/F=ihe  [_  VP'F- 
But  the  LJs  PT'F,  VP'F,  formed  by  the  line  FP  meet 
ing  the  tangents,  are  opposite  exterior  and  interior  an 
gles.  The  tangents  are  therefore  parallel  (Cor.  1,  Th.  7, 
B.  I,  Geom). 

Hence  the  theorem ;  tangents  to  the  hyperbola,  etc. 


PROPOSITION    Y  I .— THEOREM. 

The  perpendiculars  ktfall  front  the  foci  of  an  hyperbola  on 
any  tangent  line  to  the  curve,  intersect  the  tangent  on  the  circum 
ference  of  the  circle  described  on  the  major  axis  as  a  diameter. 

In  the  hyperbola  of  which  A  A' 
is  the  major  axis,  F  and  F'  the 
foci,  and  C  the  center,  take  any 
point  in  one  of  the  branches,  as 
P,  and  through  it  draw  the  tan 
gent  line  TH'.  From  the  foci  let 
fall  on  the  tangent  the  perpendic 
ulars  FH,  F'H',  draw  PF  and  PF',  and  produce  FH 
to  intersect  PF'  in  6r.  We  are  now  to  prove  that  H  and 
Hf  are  in  the  circumference  of  a  circle  of  which  AAf  is 
the  diameter. 

Draw  CH9  producing  it  to  meet  F'H'  in  Q.  Then, 
because  Pil  is  a  tangent  to  the  curve,  it  bisects  the  angle 
FPF';  therefore  the  right-angled  triangles,  FPH  and 


74  CONIC    SECTIONS. 

HPG,  being  mutually  equiangular,  and  having  the  side 
PH  common,  are  equal.  "Whence,  FII—HG  and  PF= 
PG.  But,  by  the  definition  of  the  hyperbola,  FfP—PF 
=AAf;  }±QncQFfP—PG=FfG=AAf. 

Since  CH  bisects  the  sides  F'F  and  FG  of  the  triangle 
FGF',  we  have 

F'F:FC::FfG:  CH 

but        F'F=2FC;  therefore  Ff G=2CH=AA ' 

If  then  with  C  as  a  center  and  CA  as  a  radius,  a  cir 
cumference  be  described,  it  will  pass  through  the  point  H. 

Again ;  the  triangles  FHC  and  Fr  CQ  are  in  all  respects 
equal ;  hence  CQ=  CH,  and  Q  is  alstf  a  point  in  the  cir 
cumference  of  the  circle  of  which  A  A'  is  the  diameter. 
Therefore,  the  right-angled  triangle  QH'H,  having  for 
its  hypotenuse  a  diameter  HQ  of  this  circle,  must  have 
the  vertex,  H'  of  its  right  angle  at  some  point  in  the  cir 
cumference. 

Hence  the  theorem;  t\e  perpendiculars  let  fall,  etc. 

PROPOSITION  VII.— THEOEEM. 

The  product  of  the  perpendiculars  let  fall  from  the  foci  of 
an  hyperbola  upon  a  tangent  to  the  curve  at  any  point,  is  equal 
to  the  square  of  the  semi-minor  axis. 

Resuming  the  figure  of  the  pre 
ceding  proposition ;  then,  since 
the  semi-minor  axis,  which  we  will 
represent  by  13,  is  a  mean  propor 
tional  between  the  distances  from 
either  focus  to  the  extremities  of 
the  major  axis,  we  are  to  prove 
that 

jB2= FA  x  FA'=FHx  F'H' 

By  the  construction,  the  triangles  FHC  and  CQFf  are 
equal;  therefore  FH=F'Q  (1) 


THE   HYPERBOLA.  75 

Multiplying  both  members  of  eq.  (1)  by  F'H'  it  be 
comes 

FH  -  F'H'=F>  Q  •  F'H'  (2) 

Again,  it  was  proved  in  the  last  proposition  that  the 

points  H,  Hf  and  Q  were  in  the  circumference  of  the  cir 

cle  described  on  A  A'  as  a  diameter;  therefore  F'H'  and 

F'A  are  secants  to  this  circumference,  and  we  have 

F'  Q  :  F'A'  :  :  F'A  :^'Hf    (Cor.,  Th.  18,  B.  EJ,  Geom). 

Whence,  F'  Q  -  F'H'=F'A'  -  F'A  (3) 

But  F'A'^FA,  F'A=FA',  and  F'Q=-FH.  Making 
these  substitutions  in  eq.  (3)  it  becomes 

FH  -  F'H'=FA  -  FA'=B2. 

Hence  the  theorem  :  the  product  of  the  perpendiculars,  etc. 

Cor.  1.  The  triangles  PFH,  PF'H'  are  similar  ; 
therefore,  PF  :  PFf  :  :  FH  :  F'H' 

That  is  :  The  distances-  from  any  point  on  the  hyperbola  to 
the  foci,  are,  to  each  other,  as  the  perpendiculars  let  fall  from 
the  foci  upon  the  tangent  at  that  point. 

Cor.  2.  From  the  proportion  in  corrollary  1,  we  get 

__   PF-F'H'  =,2    PF-F'H'-FH 

FH=  --  pF,     ;  whence  FH  =- 


But  by  the  proposition,  F'H'  •  FH=&  ; 

_  2       ^2  .   pF      £2  .  pF 

therefore,     FH  =    pF,    ^^CA+PF^  tecause  F'  ® 
AA'=2CA,  and  PG=PF. 

In  like  manner  it  may  be  proved  that 

BZ  '  PF'_B\2  CA+PF) 
PF  PF 


PROPOSITION  YI II. —THEOREM. 

If  a  tangent  be  drawn  to  the  hyperbola  at  any  point,  and  al 
so  an  ordinate  to  the  major  axis  from  the  point  of  contact,  then 
will  the  semi-major  axis  be  a  mean  proportional  between  the 


76 


CONIC    SECTIONS. 


distance  from  the  center  to  the  foot  of  the  ordinate,  and  the  dis 
tance  from  the  center  to  the  intersection  of  the  tangent  with  this 
axis. 

Let  A  A'  be  the  major  axis,  F$f 
the  foci  and  C  the  center  of  the 
hyperbola.  Through  any  point, 
as  P,  taken  on  one  of  the  branch 
es,  draw  the  tangent  PT  intersect 
ing  the  axis  at  T;  also  draw  PF, 
PFf  to  the  foci,  and  the  ordinate 
PM  to  the  axis.  "We  are  now  to  prove  that 
CT :  CA.  ::  CA:  CM. 

Because  P T  bisects  the  vertical  angle  of  the  triangle 
FPF'  (Prop.  3),  it  divides  the  base  into  segments  pro 
portional  to  the  adjacent  sides  (Th.  24,  B.  II,  Geom.) 

Therefore,  F'T:  TF :  :  F'P:  PF. 

Whence,  F'T—TF:F/T+  TF: :  F'P—PF:  F'P+PF 

That  is,       2CT :  F'F :  :  AA'=2CA  :  F'P+PF 

Or,  by  inverting  the  means, 

2 CT :  2 CA  :  :  F'F :  F'P+PF  (D 

Now,  making  MF"=MF,  and  drawing  PF",  we  have, 
from  the  triangle  F'PF", 

F'F"  :  F'P+PF" :  :  F'P—PF" :  F'M—MF" 

(Prop  6,  PL  Trig.) 

But,  because  the  triangle  FPF "  is  isosceles,  and  PM  is 
a  perpendicular  from  the  vertical  angle  upon  the  base, 


therefore  the  preceding  proportion  becomes 

2CM:  F'P+PF:  :  2CA  :  F'F 
or,  2  CM :  2  CA  :  :  F'P+PF :  F'F  (2) 

Multiplying  proportions  (1)  and  (2),  term  by  term,  ob 
serving  that  the  terms  of  the  second  couplet  of  the  result 
ing  proportion  are  equal,  we  have 


THE   HYPERBOLA.  77 


Whence,  CT-CM=CA; 

which,  resolved  into  a  proportion,  becomes 

CT:CA:  :  CA  :  CM. 

Hence  the  theorem ;  if  a  tangent  be  drawn,  etc. 
SCHOLIUM. — The  property  of  the  hyperbola  demonstrated  in  this 
proposition  is  not  restricted  to  the  major  axis,  but  also  holds  true  in 
reference  to  the  minor  axis. 

The.  tangent  intersects  the  minor  axis  at  the  point  t,  and  PG  is 
an  ordinate  to  this  axis  from  the  point  of  contact.  Now,  the  simi 
lar  triangles  tCT,  TIIF,  give  the  proportion 

Ct  :  FH:\  CT  :  TH  (1) 

and  from  the  similar  triangles  PMT,  TF'IT,  we  also  have 
PM:  F'U'r.MT:  ITT  (2) 

Multiplying  proportions  (1)  and  (2),  term  by  term,  we  get 
Ct-PM  :  FH-F'H' : :  CT-MT  :  TH-H'  T  (3) 

But  FH-F'H'=B'i  (Prop.  7).  Moreover,  drawing  the  ordinate 
TV,  and  the  radius  CVoi  the  circle,  and  the  line  VJ\i,  we  have 
by  the  proposition 

CT:  CA'.-.CA  :  CM 
or,  CT:  CV::CV:  CM 

Therefore,  the  triangles  VCT  and  MCV,  having  the  angle  C 
common  and  the  sides  about  this  angle  proportional,  are  similar  (Cor. 
2,  Th.  17,  B.  II,  Geom.) ;  and  because  the  first  is  right-angled,  the 
second  is  also  right-angled,  the  right  angle  being  at  F;  hence 

VT*=CT-MT(Th.  25,  B.  II,  Geom). 

Also,  A  A'  and  HHr  are  two  chords  of  a  circle  intersecting  each 
other  at  T;  hence 

IIT-TH'=AT'TA'=  VT*  (Th.  17,  B.  Ill,  Geom). 
Substituting  for  the  terms  of  proportion  (3)  these  several  values, 
it  becomes 

Ct-PM:  B*::  VT*  :  FT'::1  : 1 
Whence,  Ct-PM=JP 

Therefore,  Ct  :  B : :  B  :  PM=  CG 

7* 


78  CONIC    SECTIONS. 

Cor.  It  has  been  proved  that  the  triangle  CVM  is  right- 
angled  at  V;  therefore,  VM  is  a  tangent  at  the  point  V 
to  the  circumference  on  A  A'  as  a  diameter,  and  Of  is  its 
sub-tangent.  But  TMia  also  the  sub-tangent  on  the  ma 
jor  axis  of  the  hyperbola  answering  to  the  tangent  PT; 
hence 

If  a  tangent  be  drawn  to  the  hyperbola  at  any  point,  and 
through  the  point  in  which  the  tangent  intersects  the  major  axis 
an  ordinate  be  drawn  to  the  circle  of  which  this  axis  is  a  diam 
eter,  the  sub-tangent  on  the  major  axis  corresponding  to  the  tan 
gent  through  the  extremity  of  this  ordinate  will  be  the  same  as 
that  of  the  tangent  to  the  hyperbola. 

PROPOSITION    IX.— THEOREM. 

In  any  hyperbola  the  square  of  the  semi-major  axis  is  to 
the  square  of  the  semi-minor  axis,  as  the  rectangle  of  the  dis 
tances  from  the  foot  of  any  ordinate  to  the  major  axis,  to  the 
'vertices  of  this  axis,  is  to  the  square  of  the  ordinate. 

Resuming  the  figure  to  Propo- 
sition  8,  the  construction  of  which 
needs  no  further  explanation,  we 
are  to  prove  that 
~CA2  rCif  :  :  A'M-AM:  PM\ 
assuming    CB  to   represent  the 
semi-minor  axis. 

From  the  similar  triangles  PMT,  TIIF  and  TH'F',  we 
derive  the  proportions 

PM-.FH:  :  MT :  TH 
PM:  F'H':  :  MT :  THr 

Whence    pjf 2 .  FH-F'Hf : :  MT2 :  TH-  TR'     (1) 
But  FII-F'H'  is  equal  to  the  square  of  the  semi-minor 
axis  (Prop.  7);  and  because  the  chords,  HHf  and  AA', 
of  the  circle  intersect  each  other  at  T,  we  have 


THE   HYPERBOLA.  79 


(Th.  IT,  B.  in,  Geom.) 

These  values  of  tlie  consequents  of  proportion  (1)  be 
ing  substituted,  it  becomes 

PM2  :  "SO2  :  :  MT*  :  W  (2) 

The  triangles  CVTand  TVM  are  similar,  and  give  the 
proportion 

:  VT*  ::  VM2  :  OF2=aT         (3) 


Comparing  proportions  (2)  and  (3),  we  find  that 


2 


PM  :      C  :  :  VM   :OA  (4) 

Because  M  Vis  a  tangent  and  MAf  a  secant  to  the  cir 
cle  A  VA'H',  we  have 

VM*=  A'  M  -  AM  (Th.  18,  B.  Ill,  Geom.) 
Placing  this  value  of  VM2  in  proportion  (4)  and  invert 
ing  the  means  of  the  resulting  proportion,  it  becomes 

PM2  :  A'M-  AM:  :  ~BC*  :~CA* 
or,  "CT  :  ~BG2  :  :  A'M-  AM:  PM* 

Hence  the  theorem  ;  in  any  hyperbola  the  square  of  the,  etc. 
Cor.  Proportion  (4)  above  may  be  put  under  the  form 

~CA2  :  ~BC2  :  :  VM*  :  PM2  (a) 

and  from  the  right-angled  triangle  CVM  we  have 


from  which,  because  CV=  CA,  we  get 


Also,  the  right-angled  triangles  CVM,  VTMare  similar; 
therefore,         CM:VM:  :  VM  :  MT 

-  Whence  VM2=  CM-  MT. 

Now,  if  in  proportion  (a)  we  place  for  VM    these  val 
ues,  successively,  we  shall  have  the  two  proportions 


C      :  :  :  CM-MT:  PM  (b) 

and  "CT  :  ~BC?  :  :  CM'-^OA2  :  PM2        (c) 


80  CONIC    SECTIONS, 

SCHOLIUM  1. — Let  us  denote  CA  by  a,  CB  by  b,  CMbj  x,  and 
PM byy;  then  A'M=x-}~ a  and  AM=x-a.  Because  CM*—  OA* 
=(CM+CA)  (CM—CA)=AMt-  Disproportion  (c),  by  substitu 
tion,  now  becomes 

a*  :  tf  : :  (^-fa)  (x— a)  :  y*.  (a') 

Whence  aY=Vx*—a?V 

or,  ay— 6V=—  a269. 

This  equation  is  called,  in  analytical  geometry,  £Ae  equation  of 
the  hyperbola  referred  to  its  center  and  axes,  in  which  x,  the  distance 
from  the  center  to  the  foot  of  any  ordinate  to  the  major  axis,  is 
called  the  abscissa.  The  equation  a2^* — &2x2= — a2&2  therefore  ex 
presses  the  relation  between  the  abscissa  and  ordinate  of  any  point 
of  the  curve. 

SCHOLIUM  2. — Let  y'  denote  tlfe  ordinate  and  x'  the  abscissa  of 
a  second  point  of  the  hyperbola;  then  we  shall  have 
aa  :  V  : :  (x'+a)  (x'—a)  :  y'* 

Comparing  this  proportion  with  proportion  (a'),  scholium  1,  we 
find 

f  :y":i  (x-f-a)  (x— a)  :  (x'+a)  (x'—a} 

That  is :  In  any  hyperbola  the  squares  of  any  two  ordinates  to  the 
major  axis  are  to  each  other,  'as  the  rectangles  of  the  corresponding 
distances  from  the  feet  of  these  ordinates  to  the  vertices  of  the  axis. 

A  similar  property  was  proved  for  the  ellipse  and  the  parabola. 

PROPOSITION   X.— THEOREM. 

The  parameter  of  the  major  axis,  or  the  latus-rectum,  of  the 
hyperbola  is  equal  to  the  double  ordinate  to  this  axis  through  the 
focus. 

Through  the  focus  F  of  the  hyperbo- 
la,  of  which  AA'  is  the  major  and  BBf 
the  minor  axis,  draw  the  chord  PPf  at 
right  angles  to  the  major  axis;  then  de- 
noting  the  parameter  by  P,  we  are  to 
prove  that 

AA'  :  BE'  :  :  BB'  :  PP'=P        (Def.  11.) 


THE    HYPERBOLA. 


81 


By  definition  6,  BC  =FA'  'FA,  and  by  proposition  9 
we  have 
~AC2  :  ~BC2  :  :  FA'  •  FA  :  PF=(JPP')2  (Cor.  Prop.  1.) 

"Whence     ~AC*  :  ~BGZ  :  :  ~B(f  :  (^PPJ 

Therefore  AC'.BG:  :JBC:  \PP'  (Th.  10,  B.  II,  Geom.) 

Multiplying  all  the  terms  of  this  last  proportion  by  2, 
it  becomes 


or,  AAf  :  BB'  :  :  BB'  :  PPf 

Hence  the  theorem  ;  the  parameter  of  the  major  axis,  etc. 


PROPOSITION   XI.—  THEOREM. 

If  from  the  vertices  of  any  two  conjugate  diameters  of  the 
hyperbola  ordinates  be  drawn  to  either  axis,  the  difference  of  the 
squares  of  these  ordinates  will  be  equal  to  the  square  of  one 
half  the  other  axis. 

Let  AA',  BBf  be  the  axes,  and 
PP',  QQf  any  two  conjugate  diam 
eters  of  the  conjugate  hyperbolas 
represented  in  the  figure.  Then, 
drawing  the  ordinates  QV,  PM, 
to  the  major  axes,  and  the  ordinates 
PS=MC,  QD=  VC,  to  the  minor 
axis,  it  is  to  be  proved  that 


and  that  ~GB*==~QV2-. 

Draw  the  tangents  PT  and  Qt,  the  first  intersecting  the 
major  axis  at  Tand  the  minor  axis  at  Tr,  and  the  second 
intersecting  the  minor  axis  at  t'  and  the  major  axis  at  t. 

ISTow,  by  proposition  8,  we  have,  with  reference  to  the 
tangent  PT, 

CT:  CA::CA:  CM, 


82  CONIC    SECTIONS. 

and  by  the  scholium  to  the  same  proposition,  we  also 
have,  with  ference  to  the  tangent  Qt  to  the  conjugate 
hyperbola, 

Ct:  CA'=CA:  :  CA  :  CV 

The  first  proportion  gives  CA  =  CT-  CM,  and  the  sec 
ond  ~CA2=  Ct  •  CV, 

Whence      CT-  CM=  Ct '  CV,  which,  in  the  form  of  a 
proportion,  becomes 

CM  i  CV::  Ct :  CT  (l) 

From  the  similar  triangles  tCQ,  CTP,  we  get 

Ct:CT::QC:PT  (2) 

and  from  the  triangles  CQV,  TPM 

QC:  PT:  :  CV :  TM  (3) 

Comparing  proportions  (1),  (2)  and  (3),  it  is  seen  that 

CM  i  CV:  :  CV:  TM 

"Whence     CV2=  CM-  TM;  but  TM=  CM—CT; 
Therefore          ~CV**=  CM2—  CT-  CM. 


And  because     CT  -  CM=  CA  (Prop.  8),  we  have 


or,  ~CA2=CM2—  ~CV2 

Again  we  have 

CT  :  CB::  CB:  PM    (Scho.,  Prop.  8) 
and         Ct'  :  CB  :  :  CB  :  CD=QV    (Prop.  8) 
Whence      CT'  -  PM=  Ct'  -  Q  V,    which,  resolved  into  a 
proportion,  becomes 

PM:  QV:  :  Ct'  :  CTf  (4) 

From  the  similar  triangles,  TfCP,  Ct'Q,  we  get 

Ct'  :  CTf  :  :  t'Q:  CP  (5) 

And  from  the  triangles  t'DQ,  CPM,  we  also  get 

t'Q:  CP:  :  t'D  :  PM  (6) 

From  proportions  (4),  (5)  and  (6)  we  deduce 


THE   HYPERBOLA. 


83 


PMi 


Whence        PM  ==  Q  V-  t'D;  but  t'D=  £      -Ctr 
therefore,     PM2=QV2—  Ctf  -  QV=~QV*—  ^ 
And  because     Ctf  •  OD=  CB2  (Prop.  8)  we  "  ave 


or  CB2=QV2—PM2 

Hence  the  theorem  ;  from  the  vertices  of  any  two,  etc. 
Cor.  By  corollary  to  proposition  9  we  have 
~CA2  :~CB2  :  :  O¥2—  ~OZ2  :  PM* 


In  like  manner,  in  reference  to  the  conjugate  hyperbo 
la,  we  shall  have 


:CA  :  ij 

—Ctf:  CV* 


or, 

By  composition, 


-^C      :  :~CA  : 

:  ~QV2  :  :~CA2  i~CA2+CV* 


But  by  this  proposition  we  have 

~CA2='CM2—CV3  ;  hence  7?A* 
therefore  ~GB2  :  ~QV2  :  :  "OT  :  CM 

Whence  CB:  QV::  CA:  CM 

or,  CAiCBn  CM-.  QV 


CM* 


PROPOSITION   XII.— THEOREM. 

The  difference  of  the  squares  of  any  two  conjugate  diameters 
of  an  hyperbola  is  constantly  equal  to  the  difference  of  the 
squares  of  the  axes. 

In  the  figure,  which  is  the  same 
as  that  of  the  preceding  proposi 
tion,  PPf  and  QQf  are  any  two  con 
jugate  diameters  (Def.  10).  It  is 
to  be  proved  that 

PP2—  ~QQ'2=AAf2— SB? 
By  proposition  11  we  have 


84 


CONIC    SECTIONS. 


and 


CA2=  CM  2— 
~CB'='QV2—PM 


therefore      CA2—  CB2=  CM*+ PM2—(  CV2+  Q  V2} 

Multiplying  each  member  of  this  equation  by  4,  observ 
ing  that  4iCA2—AA'  &c.,  it  becomes 

AA'2— BB'2=PP'2—  QQ*2 
Hence  the  theorem ;  the  difference  of  the  squares,  etc. 

PROPOSITION   XIII.— THEOREM. 

The  parallelogram  formed  by  drawing  tangent  lines  through 
the  vertices  of  any  two  conjugate  diameters  of  the  hyperbola  is 
equivalent  to  the  rectangle  contained  by  the  axes. 

Let  LMNO  be  a  parallelogram 
formed  by  drawing  tangent  lines 
through  the  vertices  of  the  two  con 
jugate  diameters  PPf,  QQf  of  the 
conjugate  hyperbolas  represented  in 
the  figure.  It  is  to  be  proved  that 

area  LMNO=AA'xBBf. 

We  have       CA  :  CB  :  :  CS  :  QV    (1)   (Cor.  Prop  11.) 
Also,  CT:  CA  :  :  CA  :  CS  (2)  (Prop.  8.) 

Multiplying  proportions  (1)  and  (2),  term  by  term,  omit 
ting  in  the  first  couplet  of  the  result  the  common  factor 
CA,  and  in  the  second  the  common  factor  CS,  we  find 

CT:  CB::  CA:  QV 
Whence  CT  -  Q  V=  CA  -  CB 

But  CT-  QV  measures  twice  the  area  of  the  triangle' 
CQT,  and  this  triangle  is  equivalent  to  the  half  of  the 
parallelogram  QCPL,  because  they  have  the  common  base 
QC  and  are  between  the  same  parallels  QC,  LT  (Th.  30, 
B.  I,  Geom.) 


THE    HYPERBOLA.  85 

Now  the  parallelogram  QCPL  is  one-fourth  of  the  par 
allelogram  LMNO,  and  CA  •  CB  measures  one  fourth  of 
the  rectangle  contained  "by  the  axes ;  therefore  the  paral 
lelogram  and  rectangle  are  equivalent. 

Hence  the  theorem ;  the  parallelogram  formed,  etc. 

PROPOSITION   XIV.— THEOREM. 

If  a  tangent  to  the  hyperbola  be  drawnthrough  the  vertex  of  the 
transverse  axis,  and  an  ordinate  to  any  diameter  be  drawn  from 
the,  same  point,  the  semi-diameter  will  be  a  mean  proportional 
between  the  distances,  on  the  diameter,  from  the  center  to  the  tan 
gent,  and  from  the  center  to  the  ordinate. 

Let  CA  be  the  semi-major  axis  and 
CP  any  semi-diameter  of  the  hyper- 
bola.  Draw  the  tangents  At,  PT,  the 
ordinate  AH  to  the  diameter,  and  the 
ordinate  PMto  the  major  axis.  It  is  T  AV M 
now  to  be  proved  that~OP2=  Ci  -  CH. 

"We  have  CT :  CA  :  :  CA  :  CM,  (Prop.  8) 

also  CAiCti:  CM:  OP  from  the  similar  A's  CAt,  CMP 
Multiplying  these  proportions  term   by  term,  omit 
ting  in  the  result  the  common  factor  in  the  first  couplet, 
and  also  that  in  the  second,  we  find 

CT:Ct::CA:CP  (1) 

Again  we  have 

CP'.CT: :  CH:  CA  from  the  similar  A's  CPT,  CHA. 

Proceeding  with  these  last  proportions  as  with  those 
above,  we  find 

CP^Ct::  CH:  CP 

Whence,  CP2=Ct-CH. 

Hence  the  theorem ;  if  a  tangent  to  the  hyperbola,  etc. 

Cor.  1.  From  proportion  (1)  we  get  CT-  CP=  Ct  -  CA;  but 
the  triangles  CTP,  CAt,  having  a  common  angle,  C,  are 


86  CONIC    SECTIONS. 

to  each  other  as  the  rectangles  of  the  sides  about  this  an 
gle  (Th.  23,  B.  II,  Geom.)     Therefore  ACTP=ACLi. 

Cor.  2.  If  from  the  equivalent  areas  ACTP,  A(M  we 
take  the  common  area  C  TVt  there  will  remain  A  TA  Y== 


Cor.  3.  If  we  add  to  each  of  the  triangles  TAV,  tVP, 
the  trapezoid  VAMP,  we  shall  have  area  &TMP= 
area  tAMP. 

PROPOSITION  XY.-THEOREM. 

If  through  any  point  of  an  hyperbola  there  be  drawn  a  tan 
gent,  and  an  ordinate  to  any  diameter,  the  semi-diameter  will 
be  a  mean  proportional  between  the  distances  on  the  diameter 
from  the  center  to  the  tangent,  and  from  the  center  to  the  ordi 
nate. 

Take  any  point  as  D  on  the  hy 
perbola  of  which  CA  is  the  semi- 
major  axis,  and  through  this  point  B 
draw  the  tangent  DT and  the  semi- 
diameter  CD,  also  take  any  other 
point,  as  P,  on  the  curve,  and  draw  c 
the  tangent  Pi,  the  ordinate  PIfto 
the  diameter  through  D,  and  the  ordinates  PQ  and 
DG  to  the  axis.  The  semi-diameter  CD  and  the  tangent 
Pi  intersect  each  other  at  t' '.  "We  will  now  prove  that 


Let  CB  represent  the  semi-conjugate  axis,  then  by  co 
rollary  to  proposition  9  (proportion  (b))  we  have 

~3Z2:  "CB2::  CG'TG 
and  -CA2  :  "OS2  :  :  CQ-tQ  : 

Whence    C&  '  TG  :  CQ  :tQ  :  D(?  :  ~PQ* 

but  ~D(?  :  ~PQ2  :  :~TG?  :  ~LQ\  from  the  similar  A's 
TGD,  LQP-, 


THE    HYPERBOLA.  37 


therefore         CG'TG :  CQ-tQ  :  :   TG  :  LQ         (1) 
Drawing  Dm  parallel  to  Pt  we  have  t£e  similar  A's 
j  tQP  which  give  the  proportion 

Da  :  PQ  :  :  Gm  :  Qt.  (2) 

The  A's  TGD,  LQP  also  give 

DG  :  PQ  :  :  TG  :  LQ  (3) 

From  proportions  (2)  and  (3)  we  get 

TG:LQ::Gm:Qt  (4) 

.Multiplying  proportions  (1)  and  (4)  term  by  term,  there 
results, 

CG'~TG2:  CQ-tQ'LQ:  :  ~TG2'Gm  : ~LQ2-Qt 
Dividing  the  first  and  third  terms  of  this  proportion  by 

!T6r2and  the  second  and  fourth  terms  by  Qt' LQ  it  be 
comes 

CG  :  CQ  :  :  Gm  :  LQ 

or  CG :  Gm::  CQ:  LQ  (5) 

Whence          CG  :  CG—Gm  :  :  CQ  :  CQ—LQ 
That  is  CG  :  Cm  :  :  CQ  :  CL  (6) 

Again  CT-CG=  CA*=  CQ  -  Ct,  (Prop.  8.) 

therefore  CG^:  Ct :  :  CQ  :  CT 

The  antecedents  in  this  last  proportion  and  in  propor 
tion  (6)  are  the  same,  the  consequents  are  therefore  pro 
portional,  and  we  have 

Ct:  CT::  Cm  :  CL 

We  have  also,  Cm  :  CD  :  :  Ct :  Ct'  from  the  similar 
A's  CmD,  Cttr 

And  CT:  CD::  CL  :  Off  from  the  similar  A's  CTD 
CLH 

By  the  multiplication  of  the  last  three  proportions  term 
by  term  we  find 

Ct'Cm-CT:~CD2'CT^:  Cm-Ct-CL:  CL'Ct'-CH 
Whence  CT:~CD2-CT::  CL  :  CL'Ct'-CII 
-or  l:~CD2  :  :  I  :  Ctf -CII 

therefore  ~CI?=  Ct'  -  CII 


88  CO  NIC    SECTIONS. 

Hence  the  theorem  ;  if  through  any  point  of  an,  etc. 
REMARK.  —  The  property  of  the  hyperbola  just  established  is  the 
generalization  of  that  demonstrated  in  the  preceding  proposition. 

PROPOSITION   XVI.—  THEOREM. 

The  square  of  any  semi-diameter  of  the  hyperbola  is  to  the 
square  of  its  semi-conjugate  as  the  rectangle  of  the  distances 
from  the  foot  of  any  ordinate  to  the  first  diameter,  to  the  ver 
tices  of  that  diameter,  is  to  the  square  of  the  ordinate. 

Let  PPf  and  QQ'  be  any  two 
conjugate  diameters  of  the  conju 
gate  hyperbolas  represented  in 
the  figure.  Through  any  point  as 
Cr  draw  the  tangent  G-T  inter 
secting  the  first  diameter  at  T 
and  the  second  at  Tf,  and  from 
the  same  point  draw  the  ordinates  GrH,  GrK,  to  these 
diameters. 

"We  will  now  prove  that, 

~CP2  :  ~C     :  :  PH-P'H* 


_ 

By  the  preceding  proposition  we  have   CP  =  CT-  CH 
and  multiplying  each  member  of  this  equation  by  CH  it 
~ 


becomes  CP2  -  CH=  CT-  GH 


Whence  CPj^Cff*  1  1  CT  :  Offfrom  which  by  division 
we  get  CP2  :  CH2—CP2:  :  CT  :  CH—CT=TH,     (1) 

Again  we  havel7§2=  CT'  •  C2"(Prop.  15)  and  multi 
plying  each  member  of  this  equation  by  CK  it  becomes 


Whence   CQ    :  CK    :  :  CT'  :  CK=GH  (2) 

The  similar  A's  TCT',  THGr  give  the  proportion 
CT'  :  aH:  :  CT:  TH  (3) 

Comparing  proportions  (2)  and  (3)  we  obtain 


CQ    :  CK*  ::CT:  Til  (4) 


THE    HTPERBOLA.  89 

And  by  comparing  proportions  (1)  and  (4)  we  obtain 

~CQ*  :  ~CK2  :  ~CP2  :  CH2—CP2 
or        'CP2  :  ~CQ2  :  CH2—  CP2  :  CK2=GH2 
But  because  CP=CP'  and  ~CH2—~CP2=(CH—CP) 

(CH+CP)  =  PH-  (CH+CP)  the  last  proportion  above 
becomes  ~CP*  :  ~CQ2  :  :  PH-P'H:  GH* 

Hence  the  theorem ;  The  square  of  any  semi-diameter ',  etc. 

REMARK. — The  property  of  the  hyperbola  with  reference  to  any 
two  conjugate  diameters  just  demonstrated  is  the  same  as  that  with 
reference  to  the  axes  established  in  proposition  9. 

Cor.  If  the  ordinate  GH  be  produced  to  intersect  the 
curve  at  Gf  and  the  above  construction  and  demonstra 
tion  be  supposed  made  for  the  point  Gf  instead  of  6r,  we 
should  finally  get  the  same  proportion  as  before,  except 
the  fourth  term,  which  would  be  G'H  ;  therefore,  G'H— 
GH.  Hence  we  conclude  that 

Any  diameter  of  the  hyperbola  bisects  all  the  chords  drawn 
parallel  to  a  tangent  line  through  the  vertex  of  that  diameter. 

PROPOSITION   XYII .— T  H  E  0  R  E  M . 

The  squares  of  the  ordinates  to  any  diameter  of  the  hyper 
bola  are  to  one  another  as  the  rectangles  of  the  corresponding 
distances  from  the  feet  of  these  ordinates  to  the  vertices  of  the 
diameter. 

Resuming  the  figure  to  the 
proposition  which  precedes  and 
drawing  any  other  ordinate  gh  to 
the  diameter  PP',  it  is  to  be 
proved  that 
~GH2  :~gh2:'.  PH'P'H  :  Ph-Pfh 

By  the  foregoing  proposition 
we  have  two  proportions  following,  viz  : 
~CP2  :~CQ2  ::  PH-P'H^ 

CP2  :    CQ2  ::  Ph  -P'h  :  gh 
8* 


90 


CONIC    SECTIONS. 


Since  the  ratio    CP2  :    CQ2  is  common  to  these  pro 
portions  the  remaining  terms  are  proportional. 

That  is         GH2  i^h2  i-.PH  -PfH :  Ph  -P'h 
Hence  the  theorem —  The  squares  of  the  ordinates,  etc. 


PROPOSITION    XYIII.-THEOKEM. 

If  a  cone  be  cut  by  a  plane  making  an  angle  with  its  base 
greater  than  that  made  by  an  element  of  the  cone,  the  section 
will  be  an  hyperbola. 

Let  the  A's  MVN,  BVR\)Q  the 
sections  of  two  opposite  cones  by  a 
plane  through  the  common  axis,  and 
PH  a  line  in  this  section  not  pass 
ing  through  the  vertex,  and  making 
with  MN  the  \_BHN>  the  [_BMN. 
Through  this  line  pass  a  plane  at 
right  angles  to  the  first  plane,  mak- 
ing  in  the  lower  cone  the  section 
IGAGfIf;  then  will  this  section  be  one  of  the  branches 
of  an  hyperbola. 

Let  KL  and  MNloe  the  diameters  of  two  circular  sec 
tions  made  by  planes  at  right  ^  angles  to  the  axis  of  the 
cone,  and  at  F  and  JT,  the  intersections  of  these  lines 
with  BH,  erect  the  perpendiculars  FG,  HI  to  the  plane 
MVN.  FG  is  the  intersection  of  the  plane  of  the  section 
IGA  Gfl'  with  the  plane  of  the  circle  of  which  KL  is  the 
diameter  and  is  a  common  ordinate  of  the  section  and  oi 
the  circle ;  so  likewise  is  HI  a  common  ordinate  of  the 
section  and  of  the  circle  of  which  MN  is  the  diameter. 

Now  by  the  similar  A's  AFL,  AHN,  and  BFK,  BHM 
we  have 

AFiAHi-.FL'.HN  (1) 

and  BF:BH::FKiHM  (2) 

Multiplying  proportions  (1)  and  (2),  term  by  term,  we  get 


THE   HYPERBOLA.  91 

AF  -BF :  AH-BH :  :  FL-FK :  HN  -HM      (3) 

But  because  LGK  and  NIM  are  semi-circles,  FGr2  = 
FL-FK  and  ~ILI*=HN'HM.  Substituting  these  values 
for  the  terms  of  the  last  couplet  of  proportion  (3)  it  be 
comes 

AF-BF:  AH-BH-.  :  W  :  ~HI* 

If  we  denote  any  two  ordinates  of  the  corresponding 
section  of  the  opposite  cone  by  ^  and  hi  we  should  have 
in  like  manner 

Af-Bf :  Ah  -Bh  :  :  (fg)2 :  (hi)2 

If,  therefore,  AB  be  taken  as  a  diameter  of  the  curves 
cut  out  of  the  opposite  cones  by  a  plane  through  AH,  at 
right  angles  to  the  plane  VMN9  we  have  proved  that 
these  curves  possess  the  property  which  was  demonstra 
ted  in  the  preceding  proposition  to  belong  to  the  hyper 
bola. 

Hence  the  theorem ;  if  a  curve  be  cut  by  a  plane,  etc. 

ASYMPTOTES. 

DEFINITION. — An  Asymptote  to  a  curve  is  a  straight  line 
which  continually  approaches  the  curve  without  ever 
meeting  it,  or,  which  meets  it  only  at  an  infinite  distance. 

We  shall  for  the  present  assume,  what  will  be  after 
wards  proved,  that  the  diagonals  of  the  rectangle  con 
structed  by  drawing  tangent  lines  through  the  vertices  of 
the  axis  of  the  hyperbola  possess  the  property  of  asymp 
totes,  and  they  are  therefore  called  the  asymptotes  of  the 
hyperbola. 

PROPOSITION    XIX.— THEOREM. 

If  an  ordinate  to  the  transverse  axis  of  an  hyperbola  be 
produced  to  meet  the  asymptotes,  the  rectangle  of  the  segments 
into  which  it  is  divided  by  either  of  its  intersections  with  the 
curve  willbe  equivalentto  the  square  of  the  semi-conjugate  axis. 


92  CONIC    SECTIONS. 

Let  CA,  CB  be  the  semi-axes  and  Ct, 
Ct'   the   asymptotes   of  an    hyperbola. — 
Through   any  point,  as  P,  of  the   curve,    B 
draw  the  ordinate  PQ  to  the  major  axis 
and  produce  it  to  meet  the  asymptotes  at  n  c 
and  n'.     By  the  enunciation  we   are   re 
quired  to  prove  that  CB2=Pn  'Pn' 

By  Cor.  proposition  9  we  have 

~CA2  :  "OB2  :  :~C§3— ~CZa  :  ~PQ2         (1) 
And  from  the  similar  triangles  CAB',  CQn 


$, 


2 


C       :  =C£   :  :~CQ2  :  ~fyf         (2) 

Comparing  proportions  (1)  and  (2)  we  find 

~CQ2j  -CQ^-CA2  :  ~Q^2  :  ~PQ2  which  gives  by 
division   CA*  .    C*  .  :- 


or        ~SI2  :  Qn2—  ~PQ2  :  :~CQ2  :  Qn*  (3) 

From  proportions  (2)  and  (3)  we  get 


In  this  proportion  the  antecedents  are  the  same  the 
consequents  are  therefore  equal  ;  that  is 

~  ~  (Qn—PQ)=Pn-Pri 


Hence  the  theorem  ;  if  an  ordinate  to  the  major  axis,  etc. 

Cor.  Let  us  take  another  point  p  in  the  curve  and  from 
it  draw  the  ordinate  pQf  to  the  major  axis  ;  then,  as  be 
fore,  we  shall  have  CB2<=  pt  -ptf  ;  t  and  tr  being  the  in 
tersections  of  the  ordinate,  produced,  with  the  asymptotes. 

Whence  Pn  •  Pn'=pt  -ptr,  which  in  the  form  of  a  pro 
portion  becomes  Pn  :  Pt  :  :  pt'  :  Pn' 

PROPOSITION    XX  .—  T  H  E  0  R  E  M  . 

The  parallelograms  formed  by  drawing  through  the  different 
points  of  the  hyperbola  lines  parallel  to  and  meeting  the  asymp 
totes  are  equivalent  one  to  another,  and  any  one  is  equivalent  to 
one  half  of  the  rectangle  contained  by  the  semi-axes. 


IV 


P' 


THE    HYPERBOLA. 

Let  CA,  CB  be  the  semi-axes  and  Cn, 
Cn'  the  asymptotes  of  an  hyperbola.    From 
any  point,  as  P,  of  the  curve  draw  the  ordi-    J 
nate  PQ  to  the  transverse  axis,  producing  it 
to  meet  the  asymptotes  at  n,  nf,  and  through 
P  and  the  vertex  A  draw  parallels  to  the   b 
asymptotes,   forming  the    parallelograms 
PmCl,  AECD.     This  last  is  a  rhombus 
because  its  adjacent  sides  CE,  CD  are  equal,  being  the 
semi-diagonals  of  equal  rectangles. 

It  will  now  be  proved  that 

Area  PmCi  =  area  AECD=\  Eect.  AB'BC. 

By  the  proposition  which  precedes  we  have 

~CI?=Pn  -  Pn'  (1) 

And  from  the  similar  triangles  AB'E,  Pnm,  and  the 
similar  triangles  ADb',  Pin'  we  also  have 
AE :  ABf=CB  :  :  mP  :  Pn 
AD:Abf=CBi  :  Pt :  Pn' 

Multiplying  these  proportions,  term  by  term,  we  find 
AE  •  AD  :  ~CB 2 :  :  mP  •  Pt :  Pn  -  Pn' 

By  equation  (1)  the  consequents  of  this  proportion  are 
equal,  therefore  the  antecedents  are  also  equal. 

That  is,  AE  •  AD=mP  •  Pt 

If  the  first  member  of  this  equation  be  multiplied  by 
sin.  [_DAE,  and  the  second  member  by  the  sine  of  the 
equal  \_mPt  it  becomes 

AE-  AD  •  sin.  DAE=mP  •  Pt  •  sin  mPt 

But  AE  -AD  •  sin  D AE  measures  the  area  of  the  rhom 
bus  AECD  and  mP  •  Pt  sin.  mPt  measures  the  area  of 
the  parallelogram  PmCt;  therefore  the  parallelogram  and 
the  rhombus  are  equivalent.  Moreover,  because  the 
A's  AEC,  ADC  are  equal,  and  the  A's  AEC,  AEBf  are 
equivalent,  it  follows  that  the  rhombus  AECD  is  equiva- 


94  CONIC    SECTIONS. 


lent  to  the  &AB'  C,  or,  to  one  half  of  the  rectangle  con 
tained  by  the  semi-axes. 

Hence  the  theorem;  the  parallelograms  formed,  etc. 

Cor.  1.  If  from  the  rhombus  AECD  and  the  parallel 
ogram  PmCtihe  common  part  be  taken,  there  will  remain 
the  parallelogram  AKtD,  equivalent  to  the  parallelogram 
PmEK,  and  if  to  each  of  these  the  curvilinear  area  AKP 
be  added,  we  shall  have 

Area  APmE=  area  APtD. 

Had  we  proceeded  in  the  same  way  with  the  parallelo 
gram  PmCt  and  any  parallelogram  other  than  AECD  we 
should  have  had  a  like  result  ;  therefore 

If  from  any  two  points  in  the  hyperbola  parallels  be  drawn 
to  each  asymptote,  the  area  bounded  by  the  parallels  to  one 
asymptote,  the  other  asymptote,  and  the  curve  will  be  equivalent 
to  the  other  area  like  bounded. 

SCHOLIUM.  —  If  the  product  AE-AD,  which  is  a  constant  quan 
tity  be  denoted  by  a,  the  distance  Cm  by  or,  and  the  distance 
mp=  Ct  by  y,  then,  by  this  proposition,  we  shall  have  the  equation 
xy=a,  which,  in  analytical  geometry,  is  called  the  equation  of  the 
hyperbola  referred  to  its  center  and  asymptotes. 

Cor.  2.  In  the  equation  xy=a,y  expresses  the  distance 
of  any  point  of  the  curve  from  the  asymptote  on  which 

x  is  estimated.     From  this  equation  we  get  y=-.    Now 

2£ 

it  is  evident  that  as  x  increases  y  decreases,  and  finally 
when  x  becomes  infinite,  y  becomes  zero.  That  is,  the 
asymptote  continually  approaches  the  hyperbola  without 
ever  meeting  it,  or  without  meeting  it  within  a  finite  dis 
tance.  "We  were,  therefore,  justified  in  assuming  that 
the  diagonals  of  the  rectangle  formed  by  the  tangents 
through  the  vertices  of  the  axes  were  asymptotes  to  the 
hyperbola. 


ANALYTICAL  GEOMETRY. 

(95) 


ANALYTICAL   GEOMETRY. 


GENERAL  DEFINITIONS  AND  REMARKS. 

Analytical  Geometry,  as  the  terms  imply,  proposes  to  in 
vestigate  geometrical  truths  by  means  of  analysis.  In  it 
the  magnitudes  under  consideration  are  represent  by  sim- 
bolg,  such  as  letters,  terms,  simple  or  combined,  and  equa 
tions  ;  and  problems  are  then  solved  and  the  properties 
and  relations  of  magnitude  established  by  processes  pure 
ly  algebraic. 

A  single  letter,  without  an  exponent,  will  aJwjjjs  be  un 
derstood  as  denoting  the  length  of  a  line ;  and  in  general, 
any  expression  of  the  first  degree  denotes  the  length  of  a  line 
and  is,  for  thin  reason,  said  to  be  linear ;  so  likewise,  an 
equation  all  of  whose  terms  are  of  the  first  degree  is  call 
ed  a  linear  equation. 

An  expression  of  the  second  degree  will  represent  the  meas 
ure  of  a  surface,  and  an  expression  of  the  third  degree  will 
represent  the  measure  of  a  volume. 

'"When  a  term  is  of  a  higher  degree  than  the  third,  a 
sufficient  number  of  its  literal  factors,  to  reduce  it  to  this 
degree,  must  be  regarded  as  numerical  or  abstract. 

The  subject  of  Analytical  Geometry  naturally  resolves 
itself  into  two  parts. 

First.  That  which  relates  to  the  solution  of  determinate 
problems;  that  is,  problems  in  which  it  is  required  to  de 
termine  certain  unknown  magnitudes  from  the  relations 
which  they  bear  to  others  that  are  known.  In  this  case 
we  must  be  able  to  express  the  relations  between  the 
known  and  unknown  magnitudes  by  independent  equa 
tions  equal  in  number  to  the  required  magnitudes. 

(96) 


GENERAL    PROPERTIES.  97 

After  having  obtained,  by  a  solution  of  the  equations 
of  the  problem,  the  algebraic  expressions  for  the  quanti 
ties  sought,  it  may  be  necessary,  or,  at  least  desirable,  to 
construct  their  values,  by  which  we  mean,  to  draw  a  geo 
metrical  figure  in  which  the  parts  represent  the  given  and 
determined  magnitudes,  and  have  to  each  other  the  rela 
tions  imposed  by  the  conditions  of  the  problem.  This  is 
called  the  construction  of  the  expression. 

This  branch  of  analytical  geometry,  which  may  be 
termed  Determinate  Geometry !,  being  of  the  least  impor 
tance,  relatively,  will  be  omitted,  after  this  reference,  in 
the  present  treatise,  and  we  shall  pass  at  once  to  division. 

Second.  That  which  has  for  its  object  to  discover  and 
discuss  the  general  properties  of  geometrical  magnitudes. 
In  this  the  magnitudes  are  represented  by  equations  ex 
pressing  relations  between  constant  quantities,  and,  either 
two  or  three  indeterminate  or  variable  quantities,  and  for 
this  reason  it  is  sometimes  called  Indeterminate  Geometry. 

GENERAL  PROPERTIES 

OF 

GEOMETRICAL    MAGNITUDES, 

CHAPTER  I. 

OF  POSITIONS  AND  STRAIGHT  LINES  IN  A  PLANE, AND  THE 
TRANSFORMATION  OF  CO-ORDINATES. 

DEFINITIONS. 

1.  Co-ordinate  Axes  are  two  straight  lines  drawn  in  a 
plane  through  any  assumed  point  and  making  with  each 
other  any  given  angle.     One  of  these  lines  is  the  axis  of 

.  abscissas  or  the  axis  of  X;  the  other  is  the  axis  of  ordinates, 
or  the  axis  of  Y,  and  their  intersection  is  the  origin  of  co 
ordinates. 

2.  Abscissas  are  distances  estimated  from  the  axis  of  Y 
on  lines  parallel  to  the  axis  of  X ;  ordinates  are  distances 

9 


98  ANALYTICAL    GEOMETRY. 

estimated  from  the  axis  of  X  on  lines  parallel  to  the  axis 
of  Y. 

3.  The  abscissa  and  ordinate  of  a  point  together  are 
called  the  co-ordinates  of  the  point. 

4.  The  co-ordinate  axes  are  said  to  be  rectangular  when 
they  are  at  right  angles  to  each  other,  otherwise  they  are 
oblique. 

5.  The  two  different  directions  in  which  distances  may 
be  estimated  from  either  t  axis,  on  lines  parallel  to  the 
other,  are  distinguished  by  the  signs  plus  and  minus. 

6.  Abscissas  are  designated  by  the  letter  x  and  ordi- 
nates  by  the  letter  y,  and  when  unaccented  they  are  called 
general  co-ordinates,  because  they  refer  to  no  particular 
one  of  the  points  under  consideration.     "When  particular 
points  are  to  be  considered  the  co-ordinates  of  one  are 
denoted  by  x'  and  y' ';  of  another  by  x"  and  y",  etc.,  which 
are  read  x  prime,  y  prime,  x  second,  y  second,  etc. 

ILLUSTKATIONS. — Through  any  point  A 
draw  the  lines  XX' ,  YYf  making  with 
each  other  any  given  angle.  Call  XX' 

the  axis  of  abscissas  and  YYr  the  axis 

X- 


of  ordinates.  A  is  the  origin  of  co-or 
dinates,  or  zero  point.  The  four  angu 
lar  spaces  into  which  the  plane  is  divi 
ded  are  named,  respectively,^^/,  second, 
third,  and.  fourth  angles.  YAX  is  the  first  angle,  YAX' 
is  the  second  angle,  'YAX1  is  the  third  angle,  and  Y' AX 
is  the  fourth  angle. 

Take  any  point,  as  P,  in  the  first  angle,  and  from  it 
draw  Pp  parallel  to  the  axis  of  Y  and  Ppr  parallel  to  the 
axis  of  X,  the  first  meeting  the  axis  of  X  at  p,  and  the 
second  the  axis  of  Fat  pf;  then  p'P=Ap  is  the  abscissa, 
and  pP=Apr  is  the  ordinate  of  the  point  P. 

Now  produce  Ppr  to  P'  making  p'Pf=p'P,  and  from 
P'  draw  a  parallel  to  the  axis  of  Y  meeting  the  axis  of  X 
at  p";  then  the  point  P'  is  in  the  second  angle,  and  p'P' 


GENERAL    PROPERTIES.  99 

*= Ap"  is  its  abscissa,  and  p"Pf=Apr  is  the  ordinate.  By 
like  constructions  we  determine  the  position  of  the  point 
P'  in  the  third  angle,  and  that  of  the  point  P'"  in  the 
fourth  angle. 

It  is  evident  that  the  abscissas  of  these  four  points  are 
numerically  equal,  as  are  likewise  their  ordinates  ;  but  if 
we  have  reference  to  the  algebraic  signs  of  the  co-ordi 
nates,  each  point  will  be  assigned  to  its  appropriate  angle 
and  will  be  completely  distinguished  from  the  others. 
Abscissas  estimated  to  the  right  of  the  axis  of  Y  are  posi 
tive  and  those  estimated  to  the  left  are  negative.  Ordinates 
estimated  from  the  axis  of  JT  upwards  are  positive,  those 
estimated  downwards  are  negative. 
We  shall  therefore  have  for  points 

In  the  1st  angle,  x  positive,  y  positive. 

"     "    2d       "      x  negative,  y  positive. 

"     "    3d       "     x  negative  y  negative. 

"     "    4th     "      x  positive  y  negative. 

From  what  precedes  we  see  that  the  position  of  a  point 
in  the  plane  of  the  co-ordinate  axis  is  fully  determined 
by  its  co-ordinates.  To  construct  this  position  we  lay  off 
on  the  axis  of  X  the  given  abscissa,  to  the  right,  or  to  the 
left  of  the  origin,  according  to  the  sign ;  also  lay  off  on 
the  axis  of  Y  the  given  ordinate,  upwards  from  the  origin 
if  the  sign  be  plus,  downwards  if  it  be  minus.  The  lines 
drawn  through  the  points  thus  found,  parallel  to  the  co 
ordinate  axes,  will  intersect  at  the  required  point  and  fix 
its  position. 

As  rectangular  co-ordinates  are  more  readily  appre 
hended  than  oblique,  and  as  discussions  and  algebraic 
expressions  are  generally  less  complicated  where  refer 
ences  are  made  to  the  former,  than  when  made  to  the 
latter,  rectangular  co-ordinates  will  be  habitually  em 
ployed  in  the  following  pages.  When  we  have  occasion 
to  use  others  it  will  be  so  stated. 


100  ANALYTICAL    GEGMETKY, 

PROPOSITION   I- 

To  find  the  equation  of  a  straight  line, 

Let  XX',  YT  be  two  rectangu-  Y 

iar  co-ordinate  axes.  A  being  the 
origin  draw  any  line  as  LfL  through 
this  point,  and  designate  the  natu 
ral  tangent  of  the  angle  LAX  by  a. 

Then  take  any  distance  on  AX 
as  AP,  and  represent  it  by  x,  and 
the  perpendicular  distance  PMy. 

Then  by  trigonometry  we  have" 

Ead  :  tan.  MAP  :;AP:  PM 

or  1  :  a  :  :  x  :  y 

Whence  y~ax  (l) 

Now  this  equation  is  general ;  that  is,  it  applies  to  any 
point  M  on  the  line  AL,  because  we  can  make  x  greater 
or  less,  and  PM  will  be  greater  or  less  in  like  proportion 
and  M  will  move  along  on  the  line  AL  as  we  move  P  on 
the  line  AX.  Because  the  point  M  will  continue  on  the 
line  AL  through  all  changes  of  x  and  y,  we  say  that  y—ax 
is  the  equation  of  the  line  AL. 

Now  let  us  diminish  x  to  0,  and  the  equation  .reduces 
to  ?/=0  at  the  same  time,  which  brings  M  to  the  point  A. 

Let  x  pass  the  line  YY',  then  AP'  becomes— #,  and 
the  corresponding  value  of  y  will  be  P'M1,  and,being  be 
low  the  line  X'  X,  will,  therefore,be  minus. 

Therefore  y=ax. 

is  the  general  equation  of  the  line  L _Z7,  extending  indefi 
nitely  in  either  direction. 

If  the  tangent  a  becomes  less,  the  line  will  incline  more 
towards  the  line  X'X.  When  a=0  the  line  will  coincide 
with  Xlt>. 

Fow  let  AP*"  fce  +&W&  a  become  —a,  then  P"'M"> 
will  correspond  to  y,  ®x&  beepmes  minm  y,  because  it  is 


STRAIGHT   LiJN'ES. 


101 


below  the  axis  &X' .     Or,  algebraically  y= — ax,  indica 
ting  some  point  JM.'"  below  the  horizontal  axis. 

It  is,  therefore,  obvious  that  y—ax  may  represent  any 
line,  as  LL',  passing  through  A  from  the  list  into  the  %d 
quadrant,  and  that  y= — ax  may  be  made  to  represent  any 
line,  as  L*I/",  passing  through  A  from  the  2d  into  the 
4th  quadrant. 

Therefore  y=±&x 

may  be  wade  to  represent  any  straight  line  passing  tJirouyh  the 
zero  point. 

In  case  we  have  — a  and  — x,  that  is,  both  a  and  x  mi 
nus  at  the  same  time,  their  product  will  be  -+(ix,  showing 
that  y  must  be  ^>fe?  by  the  rules  of  algebra. 

As  an  exercise,  Set  the  learner  examine  these  lines  and 
see  whether  they  ^correspond  to  the  equation. 

When  we  have  — a  we  must  draw  the  line  from  A  to 
the  right  and  below  AX ;  then  XAL'"  is  the  angle  whose 
natural  tangent  is  —a.  But  the  opposite  angle  X^AU'is 
the  same  in  value. 

When  we  have  — x  we  must  take  the  distance  as  AP' 
to  the  left  of  the  axis  YY7,  and  the  corresponding  line 
P'M"  is  above  XX' ,  and  therefore  plus,  as  it  ought  to  be. 

But  the  equation  of  a  straight 
line  passing  through  the  zero 
point  is  not  sufficiently  general 
for  practical  application ;  we  will 
therefore  suppose  a  line  to  pass 
in  any  direction  across  the  axis 
YY',  cutting  it  at  the  distance 
AB  or  AD  (±6)  or  b  distance 
above  or  below  the  zero  point  A, 
and  find  its  equation. 

Through  the  zero  point  A  draw  a  line,  AN,  parallel  to 
ML. 

Take  any  point  on  the  line  AX  and  through  P  draw 
9* 


102  ANALYTICAL    GEOMETRY. 

PM  parallel  to  A  Y,  then  ABMNv?\\\  be  a  parallelogram. 

Put  AP=x.      PM=y.      The  tangent  of  the  angle 
NAP=a.     Then  will  NP=ax. 

To  each  of  these  equals  add  NM=b,  then  we  shall  have 

y=ax+b 

for  the  relation  between  the  values  of  x  and  y  correspond 
ing  to  the  point  M,  and  as  M  is  any  variable  point  on  the 
line  ML  corresponding  to  the  variations  of  x,  this  equa 
tion  is  said  to  be  the  equation  of  the  line  ML. 

"When  b  is  minus  the  line  is  then  QL',  and  cuts  the  axis 
YY'  in  1),  a  point  as  far  below  A  as  B  is  above  A. 

Hence  we  perceive  that  the  equation 


may  represent  the  equation  of  any  line  in  the  plane  YAX. 

If  we  give  to  a,  x,  and  b,  their  proper  signs,  in  each 
case  of  application  we  may  write 

y=ax+b 
for  the  equation  of  any  straight  line  in  a  plane. 

Cor.  Since  the  equation  y—ax+b  truly  expresses  the 
relation  between  the  co-ordinates  of  any  point  of  the  line, 
it  follows  that  if  the  co-ordinates  x'  and  y1  of  any  partic 
ular  point  of  the  line  be  substituted  for  the  variables  x 
and  y  the  equation  must  hold  true  ;  but  if  the  co-ordinates 
x"  and  y",  of  any  point  out  of  the  line  be  substituted  for 
the  variables,  the  equation  cannot  be  true. 

"What  appears  in  the  particular  case  of  a  straight  line 
are  general  principles  which  we  thus  enunciate,  viz  : 

1st.  If  the  co-ordinates  of  a  particular  point,  in  any  line 
whatever,  be  substituted  for  the  variables  in  the  equation  of  the 
line,  the  equation  must  be  satisfied;  but  if  the  co-ordinates  of 
a  point  out  the  Ine,  be  substituted  for  the  variables  in  its  equa 
tion,  the  equation  cannot  be  satisfied. 

2d.  If  the  co-ordinates  of  any  point  be  substituted  for  the  va 
riables  in  the  equation  of  a  line,  and  the  equation  be  satisfied,  the 


STRAIGHT    LINES.  103 

point  must  be  on  the  line  ;  but  if  the  equation  be  not  satisfied  by 
the  substitution,  the  point  cannot  be  on  the  line. 

These  are  principles  of  the  highest  importance  in  ana 
lytical  geometry,  and  should  be  thoroughly  committed 
and  fully  understood  by  the  student. 

SCHOLIUM. — Instead   of  rectangular,  let   us   as-  Y 

sume  the  oblique  co-ordinate  axes  AX  and  A  Yt 
making  with  each  other  an  angle  denoted  by  m. 
Through  the  origin  draw  the  line  AP  making  with 
the  axis  of  x  the  angle  PAD=n ;  then  the  angle 
PAjy=m — n.  Take  any  point  as  P  in  the  line 
and  from  it  draw  PD*  and  PD  parallel,  respectively,  — 
to  the  axes  of  X  and  Y. 

From  the  triangle  APD  we  have  (Prop.  4,  Sec.  1,  Plane  Trig.) 
PI):  AD::  Sin.  PAD=Sm.  PAD1 

or  y  :  x::S'm.  n  :  Sin.  (m — n.) 

Whence  y=     sin*  n 


sm.m  —  n 


But  -  is  constant  for  the  same  line  and  may  be  repre- 

sin.  (ra  —  n 
sented  by  a. 

Therefore,  for  any  straight  line  passing  through  the  origin  of  a 
system  of  oblique  co-ordinate  axes  we  have,  as  before,  the  equation 

y—ax. 

And  if  we  denote  by  b  the  distance  from  the  origin  to  the  point 
at  which  a  parallel  line  cuts  the  axis  of  Y  above  or  below  the  origin 
we  shall  also  have  for  the  equation  of  this  line 


in  which  it  must  be  remembered  that  a  denotes  the  sine  of  the 
angle  that  the  line  makes  with  axis  of  x  divided  by  the  sine  of  the 
angle  it  makes  with  the  axis  of  Y. 

To  fix  in  the  minds  of  learners  a  complete  comprehension  of  the 
equation  of  a  straight  line,  we  give  the  following  practical 

EXAMPLES. 

1.  Draw  the  line  whose  equation  is         y=2x-}-3.  (1) 

Then  draw  the  line  represented  by     y=  —  x-j-2  (2) 

and  determine  where  these  two  lines  intersect. 


104 


ANALYTICAL    GEOMETKY. 


1     2 


Draw  FF  and  XX  at  right  angles, 
and  taking  any  convenient  unit  of  meas 
ure  lay  it  off  on  each  of  the  axes  from 
the  origin  in  both  positive  and  negative 
directions  a  sufficient  number  of  times. 

Equation  (1)  is  true  for  all  values  of 
x  and  y.  It  is  true  then  when  x=Q. 
But  when  x=0  the  point  on  the  line 
must  be  on  the  axis  FF. 

Whenaj=0.     y=3.  Y'' 

This  shows  that  the  line  sought  for  must  cut  FF'  at  the  point 

+3. 

The  equation  is  equally  true  when  ?/=0.     But  when  y=0,  the 

corresponding  point  on  the  line  sought  must  be  on  the  axis  XX', 

and  on  the  same  supposition  the  equation  becomes 


That  is,  midway  between  —  1  and  —  2  is  another  point  in  the 
line  which  is  represented  by  y—  2ar-f-3,  but  two  points  in  any  right 
line  must  define  the  line;  therefore  ML  is  the  line  sought. 

Taking  equation  (2)  and  making  x=Q  will  givey=2,  and  making 
?/  —  0  will  give  x=2;  therefore  MQ  must  be  the  line  whose  equation 
is  y—  —  x-\-2,  and  these  two  lines  with  the  axis  XX'  form  the  tri 
angle  LMQj  whose  base  is  3|  and  altitude  about  2J. 

But  let  the  equations  decide,  (not  about,)  but  exactly  the  posi 
tion  of  the  point  M  of  intersection. 

This  point  being  in  both  lines,  the  co-ordinates  x  and  y  corres 
ponding  to  this  point  are  the  same,  therefore  we  may  subtract  one 
equation  from  the  other,  and  the  result  will  be  a  true  equation, 


3s+l=0.     Or  x=—  J. 

Eliminating  x  from  the  two  equations  we  find  y=2^. 
2.  For  another  example  we  nequire  the  projection  of  the  line  repre 
sented  by  the  equation 

*==--  *   —2 
420 

Making  £e=0,  then  y—  —  2.     Making  y=0,  then  x=  —  840. 
Using  the  last  figure,  we  perceive  that  the  line  sought  for  must 


STRAIGHT    LINES. 


105 


pass  through  S  two  units  below  the  zero  point,  and  it  must  take 
such  a  direction  S  V  as  to  meet  the  axis  XX!  at  the  distance  of  840 
units  to  the  left  of  zero.  Hence  its  absolute  projection  is  practi 
cally  impossible. 

8.    Construct  the  line  whose  equation  is         y=2x-{-5. 

4.  Construct  the  line  whose  equation  is         y— — 3# — 3. 

5.  Construct  the  line  represented  by  2y— 3a?-j-5. 

6.  Construct  the  line  represented  by  y—4x — 3. 

The  lines  represented  by  equations  4  and  6  will  intersect  the  axis 
of  Y  at  the  same  point.  Why  ? 

7.  Construct  the  line  whose  equation  is         y=r2#-}-3. 

8.  Construct  the  line  whose  equation  is         y=. — 2x — 3. 

The  last  two  lines  intercept  a  portion  of  the  axis  of  Y  which  is 
the  base  of  an  isosceles  triangle  of  which  the  two  lines  are  the  sides. 
What  are  the  base  and  perpendicular,  and  where  the  vertex  of  the 

triangle  1 

ANS.  The  base  is  6,  the  perpendicular  1£,  vertex  on  the  axis  of  X. 

Construct  the  lines  represented  by  the  following  equations. 

9.  3a?+5y— 15=0 

11.  *+y+2=0 

12.  —  a?+y+3=0 

13.  2x- y+4=0 

PROPOSITION  II 

To  find  the  distance  between  two  given  points  in  the  plane  of 
the  co-ordinate  axis.  Also,  to  find  the  angle  made  by  the  line 
joining  the  two  given  points,  and  the  axis  of  X. 

Let  the  two  given  points  be  P 
and  §,  and  because  the  point  P  is 
said  to  be  given,  we  know  the  two 
distances 

AN=xf,  NP=yr. 
And  because   the  point    Q  is 
given  we  know  the  two  distances. 
AM=x"  and  MQ=y". 


P 

N 

^ 

Q 

R 

M: 

106  ANALYTICAL    GEOMETRY. 

Then,  AM—  AN=NM=PR=x"—  x'; 

and  MQ—MR=QR=y"—y'. 

In  the  right  angled  triangle  PRQ  we  have 

(PJR)2+(£Q)2=(PQ)2.    But  D=PQ. 
That  is   D2=(x"—  x'}2+(y"—  yj, 

Or  D=^(x"—x'f+(y"—yJ 

Thus  we  discover  that  the  distance  between  any  two 
given  points  is  equal  to  the  square  root  of  the  sum  of  the 
squares  of  the  differences  of  their  abscissas  and  ordinates. 

If  one  of  these  points  be  the  origin  or  zero  point,  then 
£r=0  and  ?/'=0,  and  we  have 


a  result  obviously  true. 

To  find  the  angle  between  PQ  and  AX. 

PR  is  drawn  parallel  to  A  X,  therefore  the  angle  sought 
is  the  same  in  value  as  the  angle  QPR. 

Designate  the  tangent  of  this  angle  by  a,  then  by  trigo 
nometry  we  have 

PR-.  RQ::  radius  :  tan.  QPR. 
That  is,       x"—xr  :  y"—yf  :  :  1  :  a. 

f—yf 

Whence  a==x"  _  xf 

In  case  #"=?/',  PQ  will  coincide  with  PR,  and  be  paral 
lel  to  A  X,  and  the  tangent  of  the  angle  will  then  be  0, 
and  this  is  shown  by  the  equation,  for  then 


x"—  x' 

In  case  x"=xf,  then  PQ  will  coincide  with  RQ  and  be 
parallel  to  A  Y,  and  tangent  a  will  be  infinite,  and  this 
too  the  equation  shows,  for  if  we  make  x"=xr  or  x"—xf 
=0,  the  equation  will  become 

y"—yf 
a—V.  __  ^_=oo 

0 


STRAIGHT    LINES.  107 

PROPOSITION   III. 

To  find  the  equation  of  a  line  drawn  through  any  given 
point. 

Let  P  be  the  given  point :  The  equation  must  be  in 
the  form 

y=ax+b  (1) 

Because  the  line  must  pass  through  the  given  point 
whose  co-ordinates  are  x'  and  ?/',  we  must  have 
y'=axr  +  b.  (2) 

Subtracting  equation  (2)  from  equation  (1)  member 
from  member,  we  have 

y — yr=a(x — xf)  (3) 

for  the  equation  sought. 

In  this  equation  a  is  indeterminate,  as  it  ought  to  be, 
because  an  infinite  number  of  straight  lines  can  be  drawn 
through  the  point  P. 

"We  may  give  to  yr  and  x'  their  numerical  values,  and 
give  any  value  whatever  to  a,  then  we  can  construct  a 
particular  line  that  will  run  through  the  given  point  P. 

Suppose  #'=2, 2/'=3,  and  make  a=4. 

Then  the  equation  will  become 
y_3=4(z—  2). 

Or  y—4,x — 5. 

This  equation  is  obviously  that  of  a  straight  line,  hence 
equation  (3)  is  of  the  required  form. 

PROPOSITION    IY. 

To  find  the  equation  of  a  line  which  passes  through  two 
given  points. 

Let  AX  and  A  Y  be  the  co-ordinate  axes,  and  P  and  Q 
the  given  points.  Denote  the  co-ordinates  of  P  by  x',  yr 
and  of  Q  by  x",  y". 

The  required  equation  must  be  of  the  form 


108 


ANALYTICAL    GEOMETKY. 


VVe  will  now  determine  such 
vrlues  for  a  and  b  as  will  cause  the 
Ime  represented  by  this  equation 
to  pass  through  the  given  points. 

As  the  line  is  to  pass  through 
the  point  P,  the  co-ordinates  x', 
yf  of  this  point  when  substituted 
for  the  variables  x,  y  must  satisfy 
the  equation,  and  we  shall  have 
yf=axf-\-b 


Q 


N 


M: 


(2) 


And  because  the  line  is  to  pass  through  the  point  §, 
whose  co-ordinates  are  x",y"  we  will  also  have 

y"=ax"+b  (3) 

Subtracting  eq.  (2)  from  eq.  (3)  member  from  member, 
we  get 

"Whence  a=y"~y'  (4) 

xu— x' 

From  eqs.  (1)  and  (2)  we  obtain  in  like  manner 

y — y'=a(x — xf]  (5) 

Substituting  for  a  in  eq.  (5)  its  value  in  eq.  (4)  we  find 

y—y'=y-^-.(x—x'}       (6) 

x' — x' 
for  the  equation  sought. 

If  we  subtract  eq.  (3)  from  eq.  (1)  member  from  mem 
ber,  and  substitute  for  a  in  the  resulting  equation  its  value 
in  eq.  (4)  we  find 

y—y"-- 


for  the  required  equation. 

By  simply  clearing  eqs.  (6)  and  (7)  of  fractions  and  re 
ducing,  it  may  be  shown  that  they  are  in  fact  but  different 
forms  of  the  same  equation. 

To  prove  that  either  of  these  equations  is  that  of  a  line 
passing  through  the  points  P  and  §,  we  have  but  to  sub- 


STRAIGHT    LINES.  109 

stitute  in  it,  for  x  and  y,  the  co-ordinates  of  these  points. 
It  will  be  found  that  when  these  substitutions  are  made 
for  either  point,  the  equation  will  be  satisfied. 

We  will  illustrate  the  use  of  these  equations  by  the  fol 
lowing 

EXAMPLES. 

1.  The  co-ordinates  of  P  are  x'=3,  2/'=4,   and  of  §, 

*»=-i,  y=3. 

"What  is  the  equation  of  the  line  that  passes  through 
these  points  ? 

Here 


And  the  equation  y  —  yr=&  '  ~~^  '  -(x  —  x')  becomes 

x  —  x' 


By  substituting  in  the  equation    y  —  ?/"—   7~7(#  —  x") 

•U  "'      *C 

we  gety  —  3=j(x+l)  ory=Jx-f  3J,  the  same  as  that  above. 

2.  Find  the  equation  of  the  straight  line  that  is  deter 
mined  by  the  points  whose  co-ordinates  are  xf—  —  4,  y'— 
—1  and  z"=4i,  V=—  V 

Ans.  y=—  ^x—  Ijf. 

3.  The  co-ordinates  of  one  point  are  z'=6,  2/'=5,  and 
of  another  they  are  x"=  —  3,  y"=S.     What  is  the  equation 
of  the  straight  line  that  passes  through  these  points  ? 

Ans. 


PROPOSITION   V. 

To  find  the  equation  of  a  straight  line  which  shall  pass 
through  a  given  point  and  make,with  a  given  line,  a  given  angle. 
The  equation  of  the  given  line  must  be  in  the  form 

v=ax+6.  (1) 

10 


110  ANALYTICAL    GEOMETEY. 

Because  the  other  line  must  pass  through  a  given  point 
its  equation  must  be  (Prop.  III.) 

y — y'=a'(x — xf).  (2) 

"We  have  now  to  determine  the  value  of  a'. 

"When  a  and  a'  are  equal,  the  two  lines  must  be  paral 
lel,  and  the  inclination  of  the  two  lines  will  be  greater  or 
less  according  to  the  relative  values  of  a  and  a'. 

Let  PQ  be  the  given  line, 
making  with  the  axis  of  JTan 
angle  whose  tangent  is  a  and 
Pit  the  other  line  which  shall 
pass  through  the  given  point  P 
and  make  with  P§,  a  given  an-  Q 


gle  QPR.     The  tangent  of  the      / 
angle  PPJTis  equal  to  a'.  ' 

Because  PRX=PQR+QPR. 

QPR=PRZ—PQR 
Tan.  §P#=tan.  (PRX—PQR.) 

As  the  angle  QPR  is  supposed  to  be  known  or  given, 
we  may  designate  its  tangent  by  m,  and  m  is  a  known 
quantity. 

Now  by  trigonometry  we  have 


m=tan.  (PRX—PQR}=~-f  .        (3) 


Whence  a' 


1  —  ma 
This  value  of  a'  put  in  eq.  (2)  gives 


for  the  equation  sought. 

Cor.  1.  "When  the  given  inclination  is  90°,  m  its  tan 

gent  is  infinite,  and  then  af—  —  _.  "We  decide  this  in  the 

a 

following  manner. 

An  infinite  quantity  cannot  be  increased  or  diminished 


STRAIGHT    LINES.  Ill 

relatively,  by  the  addition  or  subtraction  of  finite  quanti 
ties,  therefore,  on  that  supposition, 

1 — ma  — ma  a 

APPLICATION. — To  make  sure  that  we  comprehend  this 
proposition  and  its  resulting  equation,  we  give  the  fol 
lowing  example : 

The  equation  of  a  given  line  is  y=2x+6. 

Draw  another  line  that  will  in 
tersect  this  at  an  angle  of  45°  and 
pass  through  a  given  point  P, 
whose  co-ordinates  are 


Draw  the  line  MN  correspond 
ing  to  the  equation  y=2x+6.  Lo 
cate  the  point  P  from  its  given  co~ 
ordinates. 

Because  the  angle  of  intersection  is  to  be  45°,  w=l, 
and  a=2. 

Substituting  these  values  in  eq.  (4)  we  have 


Or  y=—  3z+12J. 

Constructing  the  line  MJR  corresponding  to  this  equa 
tion,  we  perceive  it  must  pass  through  P  and  make  the 
angle  NMR  45°,  as  was  required. 

The  teacher  can  propose  any  number  of  like  examples. 

Cor.  Equation  (3)  gives  the  tangent  of  the  angle  of  the 
inclination  of  any  two  lines  which  make  with  the  axis  of 
X  angles  whose  tangents  are  a  and  a'.  That  is,  we  have 
in  general  terms 

af  —  a 


I+aaf 

In  case  the  two  lines  are  parallel  m=0.     "Whence  a'=a, 
an  obvious  result. 


112  ANALYTICAL    GEOMETRY. 

In  case  the  two  lines  are  perpendicular  to  eacli  other, 
m  must  be  infinite,  and  therefore  we  must  put 


to  correspond  with  this  hypothesis,  and  this  gives 

«'~-i 

a 

a  result  found  in  Cor.  1. 

To  show  the  practical  value  of  this  equation  we  require 
the  angle  of  inclination  of  the  two  lines  represented  by 
the  equations  y=%x  —  6,  and  y=  —  x+2. 

Here  a=3  and  af=  —  1.     Whence 

-«-*• 

This  is  the  natural  tangent  of  the  angle  sought,  and  if 
we  have  not  a  table  of  natural  tangents  at  hand,  we  will 
take  the  log.  of  the  number  and  add  10  to  the  index,  then 
we  shall  have  in  the  present  example  10.301030  for  the 
log.  tangent  which  corresponds  to  63°  26'  6"  nearly. 

The  sign  of  the  tangent  determines  the  direction  in  which 
the  angles  are  estimated. 

2.  What  is  the  inclination  of  the  two  lines  whose  equa 
tion  are 


and  3y=__2£-f  6  ? 

Ans.  The  tangent  of  their  inclination  is  4f 

Log.  4.75  plus  10=10.676694. 
The  inclination  of  the  lines  is  therefore  78°  6'  5". 
3.  Find  the  equation  of  a  line  which  will  make  an  an 
gle  of  56°  with  the  line  whose  equation  is 


As  the  required  line  is  to  pass  through  no  particular 
point,  but  is  merely  to  make  a  given  angle  with  the 
known  line,  we  may  assume  it  to  pass  through  the  origin 
of  co-ordinates.  Its  equation  will  then  be  of  the  form 


STRAIGHT    LINES.  113 

y=arx.     We  must  now  determine  such  a  value  for  af  that 
the  two  lines  will  make  with  each  other  an  angle  of  56°. 

Represent  the  tangent  of  the  given  angle  hy  t;  then  "by 
corollary  (2) 


1+fa' 

Tn  the  tables  we  find  that  log.  tangent  of  56°  to  be  10. 
171013,  from  which  subtracting  10  to  reduce  it  to  the  log. 
of  the  natural  tangent  and  we  have  0.171013  for  the  log. 
of  /.  The  number  corresponding  to  this  is  1.483. 

Whence  a/~i-==  1.483 

From  which  we  find  a'= — 1,473  nearly  and  the  equa 
tion  of  the  line  making  with  the  given  line,  an  angle  of 
56°  is  therefore 

y=—  1.473z. 

PROPOSITION   VI. 

To  find  the  co-ordinates  which  will  locate  the  point  of  inter 
section  of  two  straight  lines  given  by  their  equations. 

We  have  already  done  this  in  a  particular  example  in 
Prop.  I,  and  now  we  propose  to  deduce  general  expressions 
ibr  the  same  thing. 

Let  y=ax+b    be  the  first  line. 

And  y=a'x-\-bf  be  the  second  line. 

For  their  point  of  intersection  y  and  x  in  one  equation 
will  become  the  same  as  in  the  other. 

Therefore  we  may  subtract  one  equation  from  the 
other,  and  the  result  will  be  a  true  equation. 

For  the  sake  of  perspicuity,  let  xl  and  yl  represent  the 
co-ordinates  of  the  common  point,  then  by  subtraction 
(a—af)x1 

Whence  x  .=— fc^l 
(o-O 
10* 


114  ANALYTICAL    GEOMETRY. 

EXAMPLE. 

At  what  point  will  the  lines  represented  by  the  two 
equations 

y=—  2x+l 
and  y=5x+~LQ  intersect  each  other. 

Here  a=— 2,  a'=5,  6=1,  £'=10.  Whence  3=—$,  y= 
-34. 

If  we  take  another  line  not  parallel  to  either  of  these, 
the  three  will  form  a  triangle. 

Then  if  we  locate  the  three  points  of  intersection  and 
join  them,  we  shall  have  the  triangle. 


PROPOSITION  VII. 

To  draw  a  perpendicular  from  a  given  point  to  a  given 
straight  line  and  to  find  its  length. 

Let  y=ax+b  be  the  equation  of  the  given  straight  line, 
and  x',  y'  the  co-ordinates  of  the  given  point. 

The  equation  of  the  line  which  passes  through  the  giv 
en  point  must  take  the  form 

y—y'=ar  (x—xf).     (Prop.  3.) 

And  as  this  must  be  perpendicular  to  the  given  line, 

we  must  have  a'=  —  -.     Therefore  the  equations  for  the 

two  lines  must  be 

y=ax-\-b  for  the  given  line;  (1) 

and  y  —  y'=  —  _(#  —  x'); 

CL 

1       fxf       \ 

Or        y—  —  -x+  (  -  +y'  )  for  the  perpendicular  line  (2) 
a       \a        / 


Let  xl  andf/j  represent  the  co-ordinates  of  the  point 
of  intersection  of  these  two  lines.     Then  by  Prop.  6, 


-      STRAIGHT    LINES. 


115 


»+«pW) 

a       \a        1 


,  -a  ,.- 


Or  we  may  conceive  x  and  y  to  represent  the  co-ordin 
ates  of  the  point  of  intersection,  and  eliminating  y  from 
eqs.  (1)  and  (2)  we  shall  find  x  as  above,  and  afterwards 
we  can  eliminate  y. 

Now  the  length  of  the  perpendicular  is  represented  by 


Whence 


v'C*,— x^+(yl— y'?=I>.  (Prop.  H.) 

1     (b+ax'—y'y 
+  (     a*+l     /  - 


perpendicular. 

If  we  put  u=b+axr  —  y'9  the  quantities  under  the  radi 
cal  will  become 


u 


"Whence  the  perpendicular 


EXAMPLES. 

1.  The  equation  of  a  given  line  is  y=3x  —  10,  and  the 
co-ordinates  of  a  given  point  are  x'=4  and  2/'=5. 

What  is  the  length  of  the  perpendicular  from  this  given 
point  to  the  given  straight  line  ?  Ans.  y^N/90. 

2.  The  equation  of  a  line  is  7/=  —  5x  —  15,  and  the  co 
ordinates  of  a  given  point  are  x/=4  and  y'=5. 

What  is  the  length  of  the  perpendicular  from  the  given 
point  to  the  straight  line  ?  Ans.  7.84+  . 


116  ANALYTICAL   GEOMETBY. 

PROPOSITION   VIII. 

To  find  the  equation  of  a  straight  line  which  will  bisect  the 

angle  contained  by  two  other  straight  lines. 

Let  y=ax+b  (1) 

and  y=a'x+V  (2) 

be  the  equations  of  two  straight  lines  which  intersect ; 

the  co-ordinates  of  the  point  of  intersection  are 

fb bf   \  du abf         /T\  TTT 

*i=—  7        2/i=— (Prop.  YI. 

\a — a'  I  a' — a 

"We  now  require  a  third  line  which  shall  pass  through 
the  same  point  of  intersection  and  form  such  an  angle 
with  the  axis  of  X  (the  tangent  of  which  may  be  repre 
sented  by  m)  that  this  line  will  bisect  the  angle  included 
between  the  other  two  lines.  "Whence  by  (Prop.  Y.)  the 
equation  of  the  line  sought  must  be 

y y     =m(X £j)  (3) 

in  which  we  are  to  find  the  value  of  m. 

Let  PN  represent  the  line  cor- 
responding  to  equation  (1)  PM  the 
line  whose  equation  is  (2),  and  PR 
the  line  required. 

Now  the  position  or  inclination 
of  PN  to  AX  depends  entirely  on 
the  value  of  a,  and  the  inclination 
of  PM  depends  on  a'  and  both  are    A 
independent  of  the  position  of  the  point  P. 
Now  RPN^RPX'—NPX*  and  M 

Whence  by  the  application  of  a  well  known  equation 
in  plane  trigonometry,  (Equation  (29),  p.  253  Plane  Trig.) 
we  have 

tan.  RPN=t&n.  ( TfPXr—NPZr}=  m~~a 

I+am 


And         tan.  MPR=tsm.  (MP37— 


of — m 


1    1-fa'w 


STRAIGHT   LIKES.  117 

But  by  hypothesis  these  two  angles  RPN  and  MPR 
are  to  be  equal  to  each  other.     Therefore 


"Whence  mz+—m=l.     *          (4) 

a'-k-a 

This  equation  will  give  two  values  of  m;  one  will  cor 
respond  to  the  line  PR,  and  the  other  to  a  line  at  right 
angles  to  PR. 

If  the  proper  value  rn  be  taken  from  this  equation  and 
put  in  eq.  (3),  we  shall  have  the  equation  required. 

Practically  we  had  better  let  the  equations  stand  as 
they  are,  and  substitute  the  values  of  #,  a'  x,  and  y,  cor 
responding  to  any  particular  case. 

To  illustrate  the  foregoing  proposition  we  propose  the 
following 

EXAMPLES, 

Two  lines  intersect  each  other  : 


is  the  equation  of  one  line.          0-) 
Is  tlmt  of  the  other  line,  (2) 

Find  the  equation  of  the  line  which  bisects  tjie  apgle 
contained  by  these  two  lines  : 

Here  a=—  2,  a'=4,  6=5,  &'=6. 

Whence          x  l  =  —  i,  and  y  l  =  *£* 
Thus  (3)  becomes 


And  eq.  (4)  becomes 

"Whence  m=0,1097  or  w=—  •  9.1097. 

y_y= 
(Or  y—  V  =— 


118  ANALYTICAL    GEOMETRY. 

Equation  (4)  is  that  of  the  line  required  ;  (3)  that  of  the 
line  at  right  angles  to  the  line  required.  All  will  be  ob 
vious  if  we  construct  the  lines  represented  by  the  eqs.  (1), 
(2),  (3),  and  (4). 

For  another  example,  find  the  equation  of  a  line  which 
bisects  the  angle  contained  by  the  two  lines  whose  equa 
tions  are 


Here  a=l,  a'=  —  20.     Whence  (4)  becomes 

m2—  ffm=l. 

Therefore  m=—  0.385,  or  +2.6. 
NOTE.  —  Two  straight  lines  whose  equations  are 

y=ax-{-'b    and    y'—a-^-l)' 

will  always  intersect  at  a  point  (unless  a—  a')  and  with  the  axis  of  Fform 
a  triangle.     The  area  of  such  triangle  is  expressed  by 


From  the  given  equations  we  find  the  co-ordinates  of 
the  intersection  of  the  lines  to  be 


For  the  line  bisecting  the  angle  included  between  the 
given  lines  we  have  either 

y—  234T2=—  0.385(3+  J?)  a) 

or,  y-W=2.6(*+if)  (2) 

By  transposition  and  reduction  (l)  becomes 

y=—  0.385Z+11.75  (3) 

and  (2)  becomes        #=2.6z+12.76  (4) 

The  lines  represented  by  eqs.  (3)  and  (±)  are  at  right  an 

gles  to  each  other.     The  latter  line  bisects  the  angle  in 

cluded  between  the  given  lines,  and  the  former  the  adja 

cent  or  supplemental  angle. 

3.  From  the  intersection  of  two  lines  whose  equations 

are 


STRAIGHT   LINES.  119 


4  (1) 

and  2?/=3a;+4  (2) 

A  third  line  is  drawn  making,  with  the  axis  of  -J,  an 
angle  of  30°.  Find  the  intersection  of  the  given  lines 
and  the  equation  of  the  third  line. 

(     The  co-ordinates  of  the  points  of  intersection 
Ans.-l  are  #,=  —  T\,  ^1=ff,  and  the  required  equation 

I  is?/—  fj=0,5773(z+A). 
4.  Two  lines  are  represented  by  the  equations 


and 

"What  kind  of  a  triangle  do  these  lines  form  with  the 
intercepted  portion  of  the  axis  of  F,  and  what  are  its  sides 
and  its  area  ? 

(     The  triangle  is  isosceles ;  its  base  on  the  axis 
Ans.  <  of  F  is  2,  the  other  sides  are  each  1.201+,     and 

Mts  area  0. 66+. 
5.  Two  lines  are  given  by  the  equations 

— 2£?/+3j:r= — 2J 
and  2§y— |x=4 

Required  the  equation  of  the  line  drawn  from  the  point 

whose  co-ordinates  are  ic"=3,  y^O  to  the  intersection  of 

the  given  lines  and  the  distance  between  these  two  points. 

,       f  The  equation  sought  is  y=— 0.717^+2.1523  and 

''  I     the  distance  is  ^(1.8)2+(2.52)2. 

TRANSFORMATION  OF  CO-ORDINATES. 

It  is  often  desirable  to  change  the  reference  of  points 
from  one  system  of  co-ordinate  axes  to  another  differing 
from  the  first  either  in  respect  to  the  origin  or  the  direc 
tion  of  the  axes,  or  both.  The  operation  by  which  this 
is  done  is  called  the  transformation  of  co-ordinates.  The 


120 


ANALYTICAL    GEOMETRY. 


"V"' 


system  of  co-ordinate  axes  from  which  we  pass  is  the  prim 
itive  system  and  that  to  which  we  pass  is  the  new  system. 

Let  J.  JTand  A  Y  be  the  primi 
tive  axes.  Take  any  point,  as  A', 
the  co-ordinates  of  which  referred 
to  AX  and  A  Y  are  x=a,  y=b  and 
through  it  draw  the  new  axes 
A'X,  and  A1  Y1  parallel  to  the 
primative  axes.  Then  denoting 
the  co-ordinates  of  any  point,  as 
M,  referred  to  the  primitive  axes  by  x  and  y,  and  the  co 
ordinates  of  the  same  point  referred  to  the  new  axes  by 
x'  and  y',  it  is  apparent  that 


A' 

TV 

I 

X' 

' 

Y 

A 

By  giving  to  a  and  b  suitable  signs  and  values  we  may 
place  the  new  origin  at  any  point  in  the  plane  of  the  prim 
itive  axes  and  the  above  formulas  are  those  for  passing 
from  one  system  of  axes  to  a  system  of  parallel  axes  hav 
ing  a  different  origin. 

The  formulas  for  the  transformation  of  co-ordinates 
must  express  the  values  of  the  primitive  co-ordinates  of 
points  in  terms  of  the  new  co-ordinates  and  those  quanti 
ties  which  fix  the  position  of  the  new  in  respect  to  the 
primitive  axes. 

PROPOSITION    IX. 

To  find  the  formulas  for  passing  from  a  system  of  rectangu 
lar  to  a  system  of  oblique  co-ordinates  from  a  different  origin. 

Let  AX,  A  Fbe  the  primitive  axes  and  A'X,  A'  Y'  the 
new  axes.  Through  any  point  as  M  draw  MP'  parallel 
to  A'  Y'  and  MP  perpendicular  to  AX.  Then  A'Pf  is 
the  new  abscissa,  P'M  the  new  ordinate  of  the  point  M, 
and  AP  and  PM  are  respectively  the  primitive  abscissa 
and  ordinate  of  the  same  point. 


STRAIGHT    LINES. 


121 


Let  AB=a,   BA'=b,  AP=x,   y 
PM=y,AfP'=x',  P'M=yf  the  an 
gle    X'A'Xv=m,   and  the    angle 
YfAfA"=n.     ~Now  by  trigonome 
try  we  have 

AfK=x'Qos.m,KP'=LH=x'  sin.  m 
P'H=KL=yf  cos.  n.  "A 

And  MH=yf  sin,  n. 

Whence  x=a+x'  cos.m+y'coa.n,y—b-}-x'  sin.  w-f  ?/'  sin.w, 
the  formulas  required. 

SCHOLIUM.  —  In  case  the  two  systems  have  the  same  origin,  we 
merely  suppress  a  and  b,  and  then  the  formulas  required  are 


x=x'  cos. 


cos.  »i.   y=x'  sin.  m-{-yf  sin.  n. 


PROPOSITION   X. 

To  fold  the  formulas  for  passing  from  a  system  of  oblique  co 
ordinates  to  a  system  of  rectangular  co-ordinates,  the  origin  be 
ing  the  same. 

Take  the  formulas  of  the  last  problem 

x—x*  cos.  m+yr  cos.  ft,  y=xf  sin.  m-\-y'  sin.  ft. 
We  now  regard  the  oblique  as  the  primitive  axes,  and 
require  the  corresponding  values  on  the  rectangular  axes. 
That  is,  we  require  the  values  of  xf  and  y' .  If  we  multi 
ply  the  first  by  sin.  ft,  and  the  second  by  cos.  ft,  and  sub 
tract  their  products,  y'  will  be  eliminated,  and  if  x'  be 
eliminated  in  a  similar  manner,  we  shall  obtain 

f__x  sin.  ft — y  cos.  ft  ,_y  cos.  m — x  sin  m 

sin.  (ft — m)  sin. (ft — m) 

SCHOLIUM. — If  the  zero  point  be  changed  at  the  same  time  in 
reference  to  the  oblique  system,  we  shall  have 

x  sin.  n — y  cos.  n          ,_        i  \_J/  cos.ra — x  sin.  m 


x'=a+' 


sn. 


w — m} 


We  will  close  this  subject  by  the  following 
11 


122  ANALYTICAL    GEOMETRT. 

EXAMPLE. 

The  equation  of  a  line  referred  to  rectangular  co-ordi 
nates  is 

y=a'x+bf. 

Change  it  to  a  system  of  oblique  co-ordinates  having 
the  same  zero  point. 

Substituting  for  x  and  y  their  values  as  above,  we  have 

xf  sin.  m+y  sin.  n=a'(xcoa.m-)-yf  cos.  n]+br. 
Eeducing 

,__(«'  cos.  m — sin.  m)x',  b' 

sin.  n — a'  cos.  m        sin.  n — a1  cos.  m 

POLAR    CO-ORDINATES. 

There  are  other  methods  by  which  the  relative  posi 
tions  of  points  in  a  plane  may  be  analytically  established 
than  that  of  referring  them  to  two  rectilinear  axes  inter 
secting  each  other  under  a  given  angle. 

For  example,  suppose  the  line 
AB  to  revolve  in  a  plane  about 
the  point  A.  If  the  angle  that 
this  line  makes  with  a  fixed  line 
passing  through  A  be  known,  and 
also  the  length  of  AB,  it  is  evident 
that  the  extremity  B  of  this  line 


will  be  determined,  and  that  there    A!  X 

is  no  point  whatever  in  the  plane  the  position  of  which 
may  not  be  assigned  by  giving  to  AB  and  the  angle 
which  it  makes  with  the  fixed  line  appropriate  values. 

The  variable  distance  AB  is  called  the  radius  vector,  the 
angle  that  it  makes  with  the  fixed  line  the  variable  angle  and 
the  point  A  about  which  the  radius  vector  turns,  the  pole. 
The  radius  vector  and  the  variable  angle  together  consti 
tute  a  system  of  polar  co-ordinates. 


STRAIGHT    LINES. 


123 


Denote  variable  angle  BAD  by  v,  the  radius  vector  by 
r  and  by  x  and  y,  the  co-ordinates  of  B  referred  to  the 
rectangular  axes  A X,  A  Y;  then  by  trigonometry  we 
have 

x—r  cos.  v  and  y=r  sin.  v. 

Now  from  the  first  of  these  we  have  r= — (v  may  re- 
cos,  v 

volve  all  the  way  round  the  pole),  and  as  x  and  cos.  v  are 
both  positive  and  both  negative  at  the  same  time,  that  is, 
both  positive  in  the  first  and  fourth  quadrants,  and  both 
negative  in  the  second  and  third  quadrants,  therefore  r 
will  always  be  positive. 

Consequently,  should  a  negative  radius  appear  in  any 
equation,  we  must  infer  that  some  incompatible  conditions 
have  been  admitted  into  the  equation. 


PROPOSITION    XI. 

To  find  the  formulas  for  changing  the  reference  of  points  from 
a  system  of  rectangular  co-ordinate  axes  to  a  system  of  polar 
co-ordinates. 

Let  A'X,  A'  Y  be  the  co-  y 
ordmate  axes,  A.  the  pole  AB 
the  radius  vector  of  any  point, 
and  AD  parallel  to  A'X  the 
fixed  line  from  which  the  va 
riable  angle  is  estimated.   De 
note    the  co-ordinates   A'E, 
AJEof  the  pole  by  a  and  b  and  A.' 
the  radius  vector  AB  by  r. 


D 


EC  X 

Draw  B  C  perpendicular  to 
A'X;  then  is  A1  C=x  the  abscissa,  and  BC=y  the  ordi- 
nate  of  the  point  B.  From  the  figure  we  have 

ArC=A'E+EC=AfE+AF=A'E+AB  cos.v 
and  BC=BF+FC=BF+AE=AE+AB  sin.  v 


124 

Whence 


ANALYTICAL    GEOMETKY. 


x=a-\-r  cos.  v 

y=b-\-r  sin.  v. 

SCHOLIUM. — If  instead  of  estimating  the  variable  angle  from  the 
line  AD,  which  is  parallel  to  the  axis  A'X,  we  estimate  it  from  the 
line  AH  which  makes  with  the  axis  the  given  angle  HAD=m  we 
shall  have 

x=a-{-r  cos.  (v-\-m) 
y=b-\-r  sin.  (x-\-ni) 


CHAPTER  II. 
THE   CIRCLE. 

LINES  OF  THE  SECOND  ORDER. 

Straight  lines  can  be  represented  by  equations  of  the 
first  degree,  and  they  are  therefore  called  lines  of  the  first 
order.  The  circumference  of  a  circle,  and  all  the  conic 
sections,  are  lines  of  the  second  order,  because  the  equa 
tions  which  represent  them  are  of  the  second  degree. 


PEOPOSITION  I. 
To  find  the  equation  of  a  circle. 

Let  the  origin  be  the  center  of 
the  circle.  Draw  AM  to  any 
point  in  the  circumference,  and  let 
fall  MP  perpendicular  to  the  axis 
of  X.  Put  AP=x,  PM=y  and 
AM—R. 

Then  the  right  angled  triangle 
APM  gives 


and  this  is  the  equation  of  the  circle  when  the  zero  point 
is  the  center. 


THE    CIECLE. 


125 


When  y=0,  xz=R*,  or  ±x=R,  that  is,  P  is  at  X or  J/. 
When  £=0,  y2=R2,  or  dby=R,  showing  that  Jf  on  the 
circumference  is  then  at  Y  or  Y". 

When  x  is  positive,  then  P  is  on  the  right  of  the  axis 
of  Y,  and  when  negative,  P  is  on  the  left  of  that  axis,  or 
between  A  and  A1 '. 

When  we  make  radius  unity,  as  we  often  do  in  trigo 
nometry,  then  x*+y2=l,  and  then  giving  to  x  or  y  any 
value  plus  or  minus  within  the  limit  of  unity,  the  equation 
will  give  us  the  corresponding  value  of  the  other  letter. 

In  trigonometry  y  is  called  the  sine  of  the  arc  XM,  and  x 
its  cosine. 

Hence  in  trigonometry  we  have  sin.2+cos.2=l. 

Now  if  we  remove  the  origin  to  A'  and  call  the  distance 
AfP=x,ihen  AP=x — R,  and  the  triangle  APM  gives 
(z~ RJ+y^R*. 

Whence  y?=2Rx—x*. 

This  is  the  equation  of  the  circle,  when  the  origin  is  on 
the  circumference. 

When  x=Q,y=Q  at  the  same  time.  When  x  is  greater 
than  2jR,  y  becomes  imaginary,  showing  that  such  an  hy 
pothesis  is  inconsistent  with  the  existence  of  a  point  in  the  cir 
cumference  of  the  circle. 

There  is  still  a  more  general  equation  of  the  circle 
when  the  zero  point  is  neither  at  the  center  nor  in  the 
circumference. 

The  figure  will  fully  illustrate. 

Let  AB=c,  BC=b.    Put  AP  Y 
=x,  or  AP'=x,  and  PM  or  Pf 
M'"=y,  CM,  CM',  &c.  each=J2. 

In  the  circle  we  observe  four 
equal  right  angled-  triangles. 
The  numerical  expression  is  the 
same  for  each.  Signs  only  indi 
cate  positions. 
11* 


126  ANALYTICAL    GEOMETRY. 

Now  in  case  CDM  is  the  triangle  we  fix  upon, 
We  put  AP=x,  then  BP=  CD=(x—c), 

PM=y,  Ml}=y—CB=(y—b}. 
Whence  (x— c)2+(y— b)2=R2  (1) 

In  case  CDMr  is  the  triangle,  we  put  AP=x  and  PMr 

=y> 

Then  (X—c)2+(b—y)2=R2  (2) 

In  case  CD'M"'  is  the  triangle,  we  put  AP'=x,  P'M" 
=!/• 

Then  (C—xf+(y—'b)'2=Sz  (3) 

If  CD'M"  is  the  triangle,  we  put  FM"=y. 
Then  (c—x)2+b—y)2=R2  (4) 

Equations  (1),  (2),  (3)?  and  (4),  are  in  all  respects  numer 
ically  the  same,   for  (c — x)2—(x — c)2,  and  (b  -y)2=(y — b)2. 
Hence  we  may  take  equation  (1)  to  represent  the  general 
equation  of  the  circle  referred  to  rectangular  co-ordinates. 
The  equation        (x—c)2+(y—b)2=R2  (1) 

includes  all  the  others  by  attributing  proper  values  and 
signs  to  c  and  b. 

If  we  suppose  both  c  and  b  equal  0,  it  transfers  the  zero 
point  to  the  center  of  the  circle,  and  the  equation  becomes 

x2+y2=2^ 

To  find  where  the  circle  cuts  the  axis  of  X  we  must 
makey=0.     This  reduces  the  general  equation  (1)  to 
(x—c)2+b2=R2. 

Or  (X—c)2=R2—b2. 

~Now  if  b  is  numerically  greater  than  J£,  the  first  mem 
ber  being  a  square,  (and  therefore  positive,)  must  be  equal 
to  a  negative  quantity,  which  is  impossible, — showing 
that  in  that  case  the  circle  does  not  meet  or  cut  the  axis 
of  JT,  and  this  is  obvious  from  the  figure. 

In  case  6=jR,  then  (x — c)2=0,  or  x=c,  showing  that  the 


THE    CIRCLE 


127 


circle  would  then  touch  the  axis  of  X.    If  we  make  #=0, 
eq.  (1)  becomes 


Or 

This  equation  shows  that  if  c  is  greater  than  R,  the 
circle  does  not  cut  the  axis  of  F,  and  this  is  also  obvious 
from  the  figure. 

If  c  be  less  than  R,  the  second  member  is  positive  in 
value,  and 


showing  that  if*  the  circumference  cut  the  axis  at  all,  it 
must  be  in  two  points,  as  at  Jf",  M"'  . 


PROPOSITION  II. 

The  supplementary  chords  in  the  circle  are  perpendicular  to 
each  other, 

DEFINITION. — Two  lines  drawn,  one  through  each  ex 
tremity  of  any  diameter  of  a  curve,  and  which  intersect 
the  curve  in  the  same  point,  are  called  supplementary 
chords. 

That  is,  the  chord  of  an  arc,  and  the  chord  of  its  sup 
plement. 

In  common  geometry  this  proposition  is  enunciated 
thus: 

All  angles  in  a  semi-circk  are  right  angles. 

The  equation  of  a  straight 

line  which  will  pass  through 

the  given  point  B,  must  be  of 

the  form  (Prop.  HI.  Chap.  I.) 

y-y'=a(x—x').       (1) 

The  equation  of  a  straight 
line  which  will  pass  through  the  given  point  JT,  must  be 
of  the  form  y—yr=a'(x—x').  (2) 


128  ANALYTICAL    GEOMETRY. 

At  the  point  B,  y'=0,    and    z'=— E,   or 
Therefore  eq.  (1)  becomes 

y=a(x+K).  (3) 

And  for  like  reason  eq.  (2)  becomes 

y=a'(x— ll\  (4) 

For  the  point  in  which  these  lines  intersect  x  and  y  in 
eq.  (8)  are  the  same  as  x  and  y  in  eq.  (4) ;  hence,  these 
equations  may  be  multiplied  together  under  this  sup 
position,  and  the  result  will  be  a  true  equation.  That 

is,     . 

?/3=aa'(z2— jR2).  (5) 

But  as  the  point  of  intersection  must  be  on  the  curve, 
by  hypothesis,  therefore,  x  and  y  must  conform  to  the  fol 
lowing  equation : 

y*+x*=R*.     Or  y*=— l(x*— R*}.        (6) 

Whence  aa'= — 1',  oraa'+l-j-O. 

This  last  equation  shows  that  the  two  lines  are  perpen 
dicular  to  each  other,  as  proved  by  (Cor.  2,  Prop.  5., 
Chap.  1.) 

Because  a  and  af  are  indeterminate,  we  conclude  that 
an  infinite  number  of  supplemental  chords  may  be  drawn 
in  the  semi-circle,  which  is  obviously  true. 

PROPOSTION  III. 

To  find  the  equation  of  a  line  tangent  to  the  circumference 
of  a  circle  at  a  given  point. 

Let  C  be  the  center  of  the  cir 
cle,  P  the  point  of  tangency,  and 
Q  a  point  assumed  at  pleasure  in 
the  circumference. 

Denote  the  co-ordinates  of  P 
by  z',  y',  and  those  of  Q9  by  of',  y",  f~  c  I  \ 

The  equation  of  a  line  passing 
through  two  points  whose  co-or- 


THE    CIRCLE.  129 

dinates  are  x',  y'  and  x",  y"  is  of  the  form  (Prop.  4, 
Chap.  1). 

y'x-.  (1) 


•JO  """"*£ 

We  are  to  introduce  in  this  equation,  first,  the  condi 
tion  that  the  points  P  and  Q  are  in  the  circumference  of 
the  circle,  which  will  make  the  line  a  secant  line,  and 
then  the  further  condition  that  the  point  Q  shall  coincide 
with  the  point  P,  which  will  cause  the  secant  line  to  be 
come  the  required  tangent  line. 

Because  the  points  P  and  Q  are  in  the  circumference 
of  the  circle,  we  have 

xn+yn=%* 

and  x"2+y"2=IP 

Whence  by  subtraction  and  factoring, 

(x'+x")  (x'—x")+(y'+y")  (y'—y")=0       (2) 
from  which  we  find 

y'—y"        x'+x" 


This  value  of  ^    ^  substituted  in  equation  (1)  gives  us 

X        X 

for  the  equation  of  the  secant  line, 


"Now,  if  we  suppose  this  line  to«  turn  about  the  point  P 
until  Q  unites  with  P,  we  shall  have  x"=xr  and  y"=yf, 
and  the  secant  line  will  become  a  tangent  to  the  circum 
ference  at  the  point  P. 

Under  this  supposition  eq.  (3)  becomes 

y-y'=-x-r  (x-x1),  (4) 

y 

x' 
in  which  _  is  the  value  of  the  tangent  of  the  angle 

which  the  tangent  line  makes  with  axis  of  X. 

I 


130 


ANALYTICAL    GEOMETRY. 


By  clearing  this  equation  of  fractions,  and  substituting 
for  xf2+yf2  its  value,  JR\  we  have  finally  lor  the  equation 
of  the  tangent  line, 

yy'+xxf=R2.  (5) 

This  is  the  general  equation  of  a  tangent  line  ;  £',2/', 
are  the  co-ordinates  of  the  tangent  point,  and  #,  y,  the 
co-ordinates  of  any  other  point  in  the  line. 

SCHOLIUM  1.  —  For  the  point  in  which 
the  tangent  line  cuts  the  axis  of  X,  we 
make^'=:0,  then 


Q 


For  the  point  in  which  it  meets  the 
axis  of  J",  we  make  x'=Q,  and 


SCHOLIUM  2.  —  A  line  is  said  to  be  normal  to  a  curve  when  it  is 
perpendicular  to  the  tangent  line  at  the  point  of  contact. 

Join  A,  Pj  and  if  APT  is  a  right  angle,  then  A  P  is  a  normal, 
and  AB,  a  portion  of  the  axis  of  X  under  it,  is  called  the  sub 
normal.  The  line  BT  under  the  tangent  is  called  the  subtangent. 

Let  us  now  discover  whether  APT  is  or  is  not  a  right  angle. 

Put  a'=  the  tangent  of  the  angle  PAT,  then  by  trigonometry 


But 


Whence 


a=- 


aa=— 


Or 


Eq.  (6) 

1 


Therefore  AP  is  at  right  angles  to  PT.  (Prop.  5.  Chap.  1.) 
That  is,  a  tangent  line  to  the  circumference  of  a  circle  at  any  point 
is  perpendicular  to  the  radius  drawn  to  that  point. 

SCHOLIUM  3. — Admitting  the  principle,  which  is  a  well-known 
truth  of  elementary  geometry,  demonstrated  in  the  preceding  scho 
lium,  we  would  not,  in  getting  the  equation  of  a  tangent  line  to  the 


THE    CIRCLE.  131 

circle,  draw  a  line  cutting  the  curve  in 
two  points,  but  would  draw  the  tangent 
line  PT  at  once,  and  admit  that  the  angle 
APT  was  a  right  angle.  Then  it  is  clear 
that  the  angle  APB=  the  angle  PTB. 

Now  to  find  the  equation  of  the  line, 
we  let  x'  and  yr  represent  the  co-ordinates  "A- 
of  the  point  P,  and  x  and  y  the  general  co-ordinates  of  the  line, 
and  a  the  tangent  of  its  angle  with  the  axis  of  X,  then  (by  Prop 
III,  Chap.  I,)  we  have 


Now  the  triangle  APB  gives  us  the  following  expression  for  the 
tangent  of  the  angle  APB,  or  its  equal  PTB, 


This  value  of  a  put  in  the  preceding  equation,  will  give  us 
y'-y=-x-t(x'-x). 

y' 

Or  y't—yy'^—x't+xx'. 

Whence  'xx'^R*^  same  as  before. 


PROPOSITION   IY. 

To  find  the  equation  of  a  line  tangent  to  the  circumference 
of  a  circle,  which  shall  pass  through  a  given  point  without  the 
circle. 

Let  H  (see  last  figure  to  the  preceding  proposition)  be 
the  given  point,  and  x"  and  y"  its  co-ordinates,  and  x'  and 
y'  the  co-ordinates  of  the  point  of  tangency  P. 

The  equation  of  the  line  passing  through  the  two  points 
H  and  P  must  be  of  the  form 

y—y"=a(x—x")  (!) 

in  which  a=  ^     &.. 

x'— x" 

Since  PH  is  supposed  to  be  tangent  at  the  point  P, 


132  ANALYTICAL    GEOMETRY.   " 

and  x'  and  y'  are  the  co-ordinates  of  this  point,  equation 
(6)  Prop.  3,  gives  us 


. 
Placing  this  value  of  a  in  equation  (1)  we  have 


for  the  equation  sought. 

This  equation  combined  with 


which  fixes  the  point  P  on  the  circumference  will  deter 
mine  the  values  of  x'  and  ?/',  and  as  there  will  be  two 
real  values  for  each,  it  shows  that  two  tangents  can  be 
drawn  from  H,  or  from  any  point  without  the  circle, 
which  is  obviously  true. 

SCHOLIUM.     We  can  find  the  value  of  the  tangent  PT  by  means 
of  the  similar  triangles  ABP,  PBT,  which  give 
x'  :  R  :  :  yr  :  PT. 


x 

More  general  and  elegant  formulas,  applicable  to  all  the  conic 
sections,  will  be  found  in  the  calculus  for  the  normals,  subnormals, 
tangents  and  subtangents 


OF   THE    POLAR    EQUATION    OF    THE   CIRCLE. 

The  polar  equation  of  a  curve  is  the  equation  of  the 
curve  expressed  in  terms  of  polar  co-ordinates.  The 
variable  distance  from  the  pole  to  any  point  in  the  curve 
is  called  the  radius  vector,  and  the  angle  which  the  radius 
vector  makes  with  a  given  straight  line  is  called  the  vari 
able  angle. 


THE    CIRCLE 


188 


PROPOSITION   T. 

To  find  the  polar  equation  of  the  circle. 
When  the  center  is  the  pole  or  the  fixed  point,  the  equa 
tion  is 

and  the  radius  vector  It  is  then  constant. 

2s~ow  let  P  be  the  pole,  and  the 
co-ordinates  of  that  point  referred 
to  the  center  and  rectangular  axes 
be  a  and  6.  Make  PJf=r,  and 
MPJP=v  the  variable  angle;  AN 
=x  and  NM=y.  Then  (Prop.  11, 
Chap.  1.)  we  have 

x—a-\-r  cos.  v,  and  y=b+r  sin  v. 

These  values  of  x  and  y  substituted  in  eq.  (1),  (ob 
serving  that  cos.2y+sin.2tf=l,)  will  give 

?^+2(a  cos.  v+b  sin.  v)r+a?-)-b2 — j 
which  is  the  polar  equation  sought. 

SCHOLIUM  1. — P  may  be  at  any  point 

on  the  plane.     Suppose  it  at  B'.     Then  a 

= — R   and   b—Q.       Substituting    these 

values  in  the  equation,  and  it  reduces  to 

ra — ZRrcos.  v=Q. 

As  there  is  no  absolute  term,  r=0  will 
satisfy  the  equation  and  correspond  to  one 
point  in  the  curve,  and  this  is  true,  as  P 
is  supposed  to  be  in  the  curve.     Dividing  by  r,  and 
r=2R  cos.  v. 

This  value  of  r  will  be  positive  when  cos.  v.  is  positive,  and  neg 
ative  when  cos.  v  is  negative ;  but  r  being  a  radius  vector  can  never 
be  negative,  and  the  figure  shows  this,  as  r  never  passes  to  the  left 
of  B\  but  runs  into  zero  at  that  point. 

When  v=0,  cos.  v=I,  then  r—BB'.     When  v=9Q,  cos.  v=Q, 
and  r  becomes  0  at  B',  and  the  variations  of  v  from  0  to  90,  deter 
mine  all  the  points  in  the  semi-circumference  BDB'. 
12 


134 


ANALYTICAL    GEOMETRY. 


SCHOLIUM  2. — If  the  pole  be  placed  at  By  then  a=.-\-R  and  6=0, 
which  reduces  the  general  equation  to 
r=  — 2R  cos.  v. 

Here  it  is  necessary  that  cos.  v  should  be  negative  to  make  r  pos 
itive,  therefore  v  must  commence  at  90°  and  vary  to  270°  j  that  is, 
be  on  the  left  of  the  axis  of  Y  drawn  through  B,  and  this  corre 
sponds  with  the  figure. 

APPLICATION.  The  polar  equation  of  the  circle  in  its  most  gen 
eral  form  is 

r'-f  2(a  cos.  v+6  sin  v)r+a?-\- b*=R*.  (1) 

If  we  make  6=0,  it  puts  the  polar  point  somewhere  on  the  axis 
of  Xj  and  reduces  the  equation  to 

r2-j-2a  cos.  ^.r-)-a2=^2.  (2) 

Now  if  we  make  v=0,  then  will  cos. 
v=l,  and  the  lines  represented  by  ±r 
would  refer  to  the  points  X}  X,  in  the 
circle. 

This  hypothesis  reduces  the  last  equa 
tion  to 

r*+2ar=(R*— aa)  (3) 

and  this  equation  is  the  same  in  form  as  the  common  quadratic  in 
algebra,  or  in  the  same  form  as 

x*±px=q. 
Whence  x=r,         2a=±p,     and     R* — at=q 


These  results  show  us  that  if  we  describe  a  circle  with  the  radius 
Vq  ~fip2>  and  place  P  on  the  axis  of  X  at  a  distance  from  the  cen 
ter  equal  to  to  £p,  then  PX  represents  one  value  of  x,  and  PX* 
the  other.  That  is, 


Or  x=  -  Jp-^2+t=  PXr, 

and  this  is  the  common  solution. 

When  p  is  negative,  the  polar  point  is  laid  off  to  the  left  from 
the  center  at  P'. 

The  operation  refers  to  the  right  angled  triangle  APM. 


THE    CIRCLE. 


135 


=\p,    PM=  tfq,  and  AM—  |/$Hri/. 
Let  the  form  of  the  quadratic  be 

x*^ipx=  —  q. 
Then  comparing  this  with  the  polar  equation  of  the  circle,  we 

have 

2a=±p.     R^—a>=—q. 


Take  AX=.R  and  describe  a  semi- 
circle.  Take  AP=$p  and  AP'=— 
%p.  From  P  and  1  '  draw  the  lines 
PMj  and  P'M  to  touch  the  circle; 
and  draw  AM,  AM. 

Here  AP  is  the  hypotenuse  of  a 
right  angled  triangle.  In  the  first  case  AP  was  a  side. 

In  this  figure  as  in  the  other,  PM=  ^/q;  but  here  it  is  inclined 
to  the  axis  of  X;  in  the  first  figure  it  was  perpendicular  to  it. 

The  figure  thus  drawn,  we  have  PX  for  one  value  of  x,  and  PX' 
is  the  other,  which  may  be  determined  geometrically. 

If  £ca  -\-px=  —  q 

or     x=  — 


Observe  that  the  first  part  of  the  value  of  x}  is  minus,  correspond 
ing  to  a  position  from  P  to  the  left. 

If  x*  —  px=  —  q, 

we  take  P  '  for  one  extremity  of  the  line  x. 


q=,     or     x=p—  yp 

Here  the  first  part  of  the  value  of  x,  (Jp),  is  plus,  because  it  is 
laid  off  to  the  right  of  the  point  Pf. 

Because  R=  |/lpa  —  q  R  or  AM  becomes  less  and  less  as  the 
numerical  value  of  q  approaches  the  value  of  ip2.  When  these 
two  are  equal,  7?=0,  and  the  circle  becomes  a  point.  When  q  is 
greater  than  ip2,  the  circle  has  more  than  vanished,  giving  no  real 
existence  to  any  of  these  lines,  and  the  values  of  x  are  said  to  be 
imaginary. 

We  have  found  another  method  of  geometrizing  quad 
ratic  equations,  which  we  consider  well  worthy  of  notice, 
although  it  is  of  but  little  practical  utility. 


136  ANALYTICAL    GEOMETRY. 

It  will  be  remembered  that  the  equation  of  a  straight 
line  passing  through  the  origin  of  co-ordinates  is 

y=ax,  (!) 

and  that  the  general  equation  of  the  circle  is 

(x^c)2+(y^b}2=fi2.  (2) 

If  we  make  6=0,  the  center  of  the  circle  must  be  some 

where  on  the  axis  of  X. 

Let  AM  represent  a  line,  the 

equation  of  which  is  y=ax,  and 

if  we  take  a=l,  AM  will  in 

cline  45°  from  either  axis,  as  rep-     [E'A[/^    c  p|         \s  X 

resented  in  the  figure.     Hence 

?/=x,  and  making  6=0,  if  these 

two  values  be  substituted  in  eq.  (2)  and  that  equation  re 

duced,  we  shall  find 

(3) 


This  equation  has  the  common  quadratic  form. 

Equation  (1)  responds  to  any  point  in  the  straight  line 
M'M.  Equation  (2)  responds  to  any  point  in  the  circum 
ference  BMMf. 

Therefore  equation  (3)  which  results  from  the  combina 
tion  of  eqs.  (1)  and  2)?  must  respond  to  the  points  M  and 
Mf,  the  points  in  which  the  circle  cuts  the  line. 

That  is,  PM  and  P'  Mf  are  the  two  roots  of  equation 
(3),  and  when  one  is  above  the  axis  of  X,  as  in  this  figure, 
it  is  the  positive  root,  and  P'Mf  being  below  the  axis  of 
X,  it  is  the  negative  root. 

When  both  roots  of  equation  (3)  are  positive,  the  circle 
will  cut  the  line  in  two  points  above  the  axis  of  X.  When 
the  two  roots  are  minus,  the  circle  will  cut  the  line  in  two 
points  below  the  axis  of  X. 

"When  the  two  roots  of  any  equation  in  the  form  of  eq. 
(3)  are  equal  and  positive,  the  circle  will  touch  the  line 
above  the  axis  of  X.  If  the  roots  are  equal  and  negative, 


THE    CIRCLE.  137 

the  circle  will  touch  the  line  below  the  axis  of  X.  In 
case  the  roots  of  eq.  (3)  are  imaginary,  the  circle  will  not 
meet  the.  line. 

We  give  the  following  examples  for  illustration  : 
f—  %=5. 

To  determine  the  values  of  y  hy  a  geometrical  construc 
tion  of  this  kind,  we  must  make 

c=_2,     and  ^Z^=5. 

a 

Whence  .#=3.74,  the  radius  of  the  circle.  Take  any 
distance  on  the  axes  for  the  unit  of  measure,  and  set  off 
the  distance  c  on  the  axis  of  X  from  the  origin,  for  the 
center  of  the  circle  ;  to  the  right,  if  c  is  negative,  and  to 
the  left,  if  c  is  positive. 

Then   from  the   center,  with   a  radius  equal  to  R= 


,  describe  a  circumference  cutting  the  line  drawn 
midway  between  the  two  axes,  as  in  the  figure. 

In  this  example  the  center  of  the  circle  is  at  (7,  the 
distance  of  two  units  from  the  origin  A,  to  the  right. 
Then,  with  the  radius  3.74  we  described  the  circumfer 
ence,  cutting  the  line  in  M  and  M1  ',  and  we  find  by  meas 
ure  (when  the  construction  is  accurate)  that  JHP=4.44, 
the  positive  root,  and  M'Pr=  —  1.44,  the  negative  root. 

For  another  example  we  require  the  roots  of  the  following 
equation  by.  construction: 


IT.  B.  "When  the  numerals  are  too  large  in  any  equa 
tion  for  convenience,  we  can  always  reduce  them  in  the 
following  manner: 

Put  y=nz,  then  the  equation  becomes 


Or  *+'-*-«. 

n       w 


12 


138  ANALYTICAL    GEOMETRY. 

Now  let  7i=  any  number  what 
ever.     If  7i=3,  then 


Here  c=2. 

2 

"Whence 

At  the  distance  of  two  units  to 
the  left  of  the  origin,  is  the  center  of  the  circle.  We  see 
by  the  figure  that  1  is  the  positive  root,  and  —  3  the  neg 
ative  root. 

But  y=nz,    n=3,    2=1,    y=3  or  —  9. 

We  give  one  more  example. 

Construct  the  equation 


7)2  _  fS. 

Here  c=4,  and  -  _  __=—  6.          Whence  _K=2. 
2 

Using  the  same  figure  as  before,  the  center  of  the  cir 
cle  to  this  example  is  at  _D,  and  as  the  radius  is  only  2, 
the  circumference  does  not  cut  the  line  M'M,  showing 
that  the  equation  has  no  real  roots. 

We  have  said  that  this  method  of  finding  the  roots  of 
a  quadratic  was  of  little  practical  value.  The  reason  of 
this  conclusion  is  based  on  the  fact  that  it  requires  more 
labor  to  obtain  the  value  of  the  radius  of  the  circle  than 
it  does  to  find  the  roots  themselves. 

Nevertheless  this  method  is  an  interesting  and  instruct 

ive  application  of  geometry  in  the  solution  of  equations. 

When  we  find  the  polar  equation  of  the  parabola,  we  shall 

then  have  another  method  of  constructing  the  roots  of  quad 

ratics  which  will  not  require  the  extraction  of  the  square  root. 

To   facilitate   the   geometrical   solution   of    quadratic 

equations  which  we    have  thus  indicated,  the  operator 

should  provide  himself  with  an   accurately  constructed 

scale,  which  is  represented  in  the  following  figure.     It 


THE    CIECLE. 


139 


23456 


consists  of  two  lines,  or  axes, 
at  right  angles  to  each  other, 
and  another  line  drawn 
through  their  intersection  and 
making  with  them  an  angle 
of  45°.  On  the  axes,  any  con-  ( 
venient  unit,  as  the  inch,  the 
half,  or  the  fourth  of  an  inch, 
etc.,  is  laid  off  a  sufficient 
number  of  times,  to  the  right 
and  the  left,  above  and  below  the  origin,  from  which  the 
divisions  are  numbered  1,  2,  3,  etc.,  or  10,  20,  30,  etc.,  or 
.1,  .2,  .3,  etc.  To  use  this  scale,  a  piece  of  thin,  transpa 
rent  paper,  through  which  the  numbers  may  be  distinctly 
seen,  is  fastened  over  it,  and  with  the  proper  center  and 
radius  the  circumference  of  a  circle  is  described.  The 
distances  from  the  axis  of  JT  of  the  intersections  of  this 
circumference,  with  the  inclined  line  through  the  origin, 
will  be  the  roots  of  the  equation,  and  their  numerical 
values  may  be  determined  by  the  scale. 

By  removing  one  piece  of  paper  from  the  scale  and 
substituting  another,  we  are  prepared  for  the  solution  of 
another  equation,  and  so  on. 

EXAMPLES. 

1.  Given  x2+llx=80,  to  find  x.     Ans.  x=5,  or — 16. 

2.  Given  z2— 3x=28,  to  find  x.     Ans.  z=7,  or — 4. 

3.  Given  x2 — x=2,  to  find  x.     Ans.  x—2,  or — 1. 

4.  Given  x2— 12x=— 32,  to  find  x.     Ans.  z=4,  or  8. 

5.  Given  x2— 12x=— 36,  to  find  x.     Ans.  Each  value 
is  6. 

6.  Given  x2— I2x=— 38,  to  find  x.     Both  values  imag 
inary. 

7.  Given  x2+6x=— 10,  to  find  x.     Both  values  imag 
inary. 

8.  Given  2*=  81,  to  find  x.     Ans.  x=9,  or— 9. 


140  ANALYTICAL    GEOMETEY. 

For  example  8,  c=o  and  __ZL_=81; 

"Whence,  J2=9\/2. 

This  method  may  therefore  be  used  for  extracting  the 
square  root  of  numbers.  In  such  cases,  the  center  of  the 
circle  is  at  the  zero  point. 


CHAPTER 
THE     ELLIPSE. 

have  already  developed  the  properties  of  the  El 
lipse,  Parabola  and  Hyperbola  by  geometrical  processes,  and 
it  is  now  proposed  to  re-examine  these  curves,  and  de 
velop  their  properties  by  analysis. 

As  he  proceeds,  the  student  cannot  fail  to  perceive  the 
superior  beauty  and  simplicity  of  the  analytical  methods 
of  investigation;  and,  even  if  a  knowledge  of  the  conic 
sections  were  not,  as  it  is,  of  the  highest  practical  value, 
the  mental  discipline  to  be  acquired  by  this  study  would, 
of  itself,  be  a  sufficient  compensation  for  the  time  and 
labor  given  to  it. 

As  all  needful  definitions  relating  to  these  curves  have 
been  given  in  the  CONIC  SECTIONS,  we  shall  not  repeat 
them  here,  but  will  refer  those  to  whom  such  reference 
may  be  necessary  to  the  appropriate  heads  in  that  division 
of  the  work. 

PKOPOSITION    I. 

To  find  the  equation  of  the  ellipse  referred  to  its  axes  as  the 
axes  of  co-ordinates,  the  major  axis  and  the  distance  from  the 
center  to  the  focus  being  given. 

Let  AAf  be  the  major  axis,  F^F*  the  foci,  and  C  the 
center  of  an  ellipse.  Make  CJF=c  CA=A.  Take  any 


THE   ELLIPSE.  141 


point  on  the  curve,  and  from  it 
let  fall  the  perpendicular  Pt  on 
the  major  axis  ;  then,  by  our 
conventional  notation,  is  Ct—x, 


As  F'P+PF=2A,  we  may 
put  F'P=  A+z,  and  PF=  A—  z.    Then  the  two  right  an- 
gled  triangles  F'Pt,  FPt,  give  us 

(1) 

¥  (2) 

For  the  points  in  the  curve  which  cause  t  to  fall  between 
C  and  F,  we  would  have 

(c—x)2+f=(A—zy  (3) 

But  when  expanded,  there  is  no  difference  between  eqs. 
(2)  and  (3),  and  by  giving  proper  values  and  signs  to  x 
and  y,  eqs.  (1)  and  (2)  will  respond  to  any  point  in  the 
curve  as  well  as  to  the  point  P. 

Subtracting  eq.  (2)  from  eq.  (1),  member  from  member, 
and  dividing  the  resulting  equation  by  4,  we  find 

cx=Az,  or  z=c-  (4) 

A 

This  last  equation  shows  that  F'P,  the  radius  vector, 
varies  as  the  abscissa  x. 

Add  eqs.  (1)  and  (2),  member  to  member,  and  divide 
the  result  by  2,  and  we  have 


Substituting  the  value  of  zz  from  eq.  (4),  and  clearing 
of  fractions,  we  have 


Or,  A2y2+(A2—  (?)x2=A2(A2—  c2).  (5) 

conceive  the  point  P  to  move  along  describing 
the  curve,  and  when  it  comes  to  the  point  Z),  so  that  DC 
makes  a  right  angle  with  the  axis  of  JT,  the  two  triangles 
DCF'  are  right  angled  and  equal. 


142  ANALYTICAL    GEOMETRY. 

DFf  each  is  equal  to  A,  and  as  C-F,  CF',  each  is  equal  to 
c,  we  have 


It  is  customary  to  denote  D  C  half  the  minor  axis  of  the 
ellipse  by  B,  as  well  as  half  the  major  axis  by  A,  and  ad 
hering  to  this  notation 

jB2=JL2—  c2.  (6) 

Substituting  this  in  eq.  (5),  we  have  for  the  equation 
of  the  ellipse 


referred  to  its  center  for  the  origin  of  co-ordinates. 

If  we  wish  to  transfer  the  origin  of  co-ordinates  from 
the  center  of  the  ellipse  to  the  extremity  A'  of  its  major 
axis,  we  must  put 

x=  —  A+xf,    and    y=y'. 

Substituting  these  values  of  x  and  y  in  the  last  equa 
tion,  and  reducing,  we  have 


Or  without  the  primes,  we  have 


for  the  equation  of  the  ellipse  when  the  origin  is  at  the 
extremity  of  the  major  axis. 

Cor.  1.  If  it  were  possible  for  B  to  be  equal  to  A, 
then  c  must  be  equal  to  0,  as  shown  by  eq.  (6).  Or,  while 
c  has  a  value,  it  is  impossible  for  B  to  equal  A. 

If  jB=J.,  then  e=0,  and  the  equation  becomes 

A2y2+A2x2=A2A2. 
Or  y*+3*=A*, 

the  equation  of  the  circle.  Therefore  the  circle  may  be 
called  an  ellipse,  whose  eccentricity  is  zero,  or  whose  eccen 
tricity  is  infinitely  small. 


THE   ELLIPSE.  143 

Cor.  2.     To  find  where  the  curve  cuts  the  axis  of  JT, 
make  y=0  in  the  equation,  then 


showing  that  it  extends  to  equal  distances  from  the  center. 
To  find  where  the  curve  cuts  the  axis  of  F,  make  2=0, 
and  then 


Plus  B  refers  to  tha  point  D,  —  B  indicates  the  point 
directly  opposite  to  ./),  on  the  lower  side  of  the  axis  of  JT. 

Finally,  let  x  have  any  value  whatever,  less  than  A, 
then 


an  equation  showing  two  values  of  ?/,  numerically  equal, 
indicating  that  the  curve  is  symmetrical  in  respect  to  the 
axis  of  X. 

If  we  give  to  y  any  value  less  than  jB,  the  general  equa 
tion  gives 


Showing  that  the  curve  is  symmetrical  in  respect  to  the 
axis  of  Y. 

SCHOLIUM.  —  The  ordinate  which  passes  through  one  of  the  foci, 
corresponds    to    x=c.       But    A9  —  .Z?8=ica.       Hence    A3  —  ca    or 

Ay—x't=B\      Or   (J.2—  x^=B,  and  this  value  substituted  in 

TP  272* 

the  last  equation,  gives  y==  ±  --  Whence  _  is  the  measure  of 

A  A 

the  parameter  of  any  ellipse. 


PROPOSITION   II. 

Every  diameter  of  the  ellipse  is  bisected  in  the  center. 

Through  the  center  draw  the  line  DDf.  Let  x,  and  y, 
denote  the  co-ordinates  of  the  point  D,  and  x',  #',  the 
co-ordinates  of  the  point  D'. 


144  ANALYTICAL    GEOMETRY. 

The  equation  of  the  curve  is  D 


The  equation  of  a  line  passing 
through  the  center,  must  be  of  the 
form  y=ax. 

This  equation  combined  with  the 
equation  of  the  curve,  gives 

AB  aAB 


x= 


AB 


aAB 


These  equations  show  that  the  co-ordinates  of  the  point 
Z),  are  the  same  as  those  of  the  point  D' ,  except  opposite 
in  signs.  Hence  DD'  is  bisected  at  the  center. 


PROPOSITION  III. 

The  squares  of  the  ordinates  to  either  axis  of  an  ellipse  are 
to  one  another  as  the  rectangles  of  their  corresponding  abscissas. 

Let  y  be  any  ordinate,  and  x 
its  corresponding  abscissa. 
Then,  by  the  first  proposition, 
we  shall  have 


Let  yf  be  any  other  ordinate, 
and  x'  its  corresponding  abscis 
sa,  and  by  the  same  proposition  we  must  have 


Dividing  one  of  these  equations  by  the  other,  omitting 
common  factors  in  the  numerator  and  denominator  of  the 
second  member  of  the  new  equation,  we  shall  have 


THE    ELLIPSE.  145 

f  _  (2A—x)x 
yn     (2A—x')xr 

Hence,        y*  :  yl2=(2A—x)x  :  (2A—x')xf.  (1) 

By  simply  inspecting  the  figure,  we  cannot  fail  to  per 

ceive  that  (2  A  —  x),  and  x,  are  the  abscissas  corresponding 

to  the  ordinate  y,  and  (2  A  —  x')  and  x'  are  those  corres 

ponding  to  y*. 

If  we  transfer  the  origin  to  the  lower  extremity  of  the 

conjugate  axis,  the  equation  of  the  ellipse  may  be  put 

under  the  form 


and  by  a  process  in  all  respects  similar  to  the  above,  we 
prove  that  ^  .  ^  .  ;  (2£_y)y  ;  (2jB_yy. 

Therefore,  the  squares  of  the  ordinates,  etc. 

SCHOLIUM,  —  Suppose  one  of  these  ordinates,  as  y'  to  represent 
half  the  minor  axis,  that  is,  y'—B.  Then  the  corresponding  value 
of  x'  will  be  A  and  (2  A  —  z',)  will  be  A,  also.  Whence  proportion 
(1)  will  become 

y1  :  B*=(2  A—  x)x  :  A*. 

In  respect  to  the  third  term  we  perceive  that  if  A'  His  represented 
by  x,  AH  will  be  (2A  —  #),  and  if  G  is  a  point  in  the  circle,  whose 
diameter  is  A'  A}  and  GH  the  ordinate,  then 

(2A—  x*)x= 
and  the  proportion  becomes 


Or  y  :  GH=B  :  A. 

Or  A:B=GH:y=DH. 

If  a  circumference  be  described  on  the  conjugate  axis  as  a  diam 
eter,  and  an  ordinate  of  the  circle  to  this  diameter  be  denoted  by 
X  and  the  corresponding  ordinate  of  the  ellipse  by  x,  it  may  be 
shown  in  like  manner  that 


13 


146 


ANALYTICAL    GEOMETRY. 


PROPOSITION    IY. 

The  area  of  an  ellipse  is  a  mean  proportional  between  the 
areas  of  two  circles,  the  diameter  of  the  one  being  the  major 
axis,  and  of  the  other  the  minor  axis. 

On  the  major  axis  A' A  of  the 
ellipse  as  a  diameter  describe  a 
circle,  and  in  the  semicircle  A'D 
A  inscribe  a  polygon  of  any  num 
ber  of  sides.  From  the  verti- 
ces  of  the  angles  of  this  polygon 
draw  ordinates  to  the  major  axis, 
and  join  the  points  in  which  they 
intersect  the  ellipse  by  straight  lines,  thus  constructing  a 
polygon  of  the  same  number  of  sides  in  the  semi-ellipse 
A'D' A.  Take  the  origin  of  co-ordinates  at  Af,  and  de 
note  the  ordinates  BE,  CF,  etc.,  of  the  circle  by  Y,  F', 
etc.,  the  ordinates  BfE,  C'F,  etc.,  of  the  ellipse  by  y,  yr, 
etc.,  and  the  corresponding  abscissas,  which  are  common 
to  ellipse  and  circle,  by  x,  xf,  etc. 

Then  by  the  scholium  to  Prop.  3,  we  have 

TiynAiB 

and  Y'  :  y' :  :  A  :  B, 

whence  Y :  Yf :  :  y  :  y', 

from  which,  by  composition,  we  get 

Y+  Y'  :  y+y'  :Y:y::A:B 
But  the  area  of  the  trapezoid  BEFC  is  measured  by 


and  that  of  the  trapezoid  B'EFG'  by 
( \(x  — Xj  or  \y~\~ 

therefore, 

trapez.  BEFC      Y+  Y'    A 


THE    ELLIPSE.  147 

That  is,  trapez.  BEFC  :  trapez.  B'EFC'  :A:B', 
or,  in  words,  any  trapezoid  of  the  semi-circle  is  to  the  corres 
ponding  trapezoid  of  the  semi-ellipse  as  A  is  to  B. 

From  this  we  conclude  that  the  sum  of  the  trapezoids 
in  the  semi-circle  is  to  the  sum  of  the  trapezoids  in  the 
semi-ellipse  as  A  is  to  B.  But  by  making  these  trape 
zoids  indefinitely  small,  and  their  number,  therefore,  in 
definitely  great,  the  first  sum  will  become  the  area  of  the 
semi-circle  and  the  second,  the  area  of  the  semi-ellipse. 

Hence, 

Area  semi-circle  :  area  semi-ellipse  :  :  A  :  B 
or,        area  circle  :  area  ellipse  :  :  A  :  B 

That  is,          xA2  :  area  ellipse  :  :  A  :  B 

"Whence,         area  ellipse=  "*!.  _  !  __  __  TrA.B 

_/± 

But  TrA.B  is  a  mean  proportional  between  nAz  and 


Hence  ;   The  area  of  an  ellipse  is  a  mean  proportional,  etc. 

SCHOLIUM.  —  Hence  the  common  rule  in  mensuration  to  find  the 
area  of  an  ellipse. 

RULE.  —  Multiply  the  semi-major  and  semi-minor  axes  together, 
and  multiply  that  product  ty  3.1416. 

PROPOSITION  Y. 

To  find  the  product  of  the  tangents  of  the  angles  that  two 
supplementary  chords  through  the  vertices  of  the  transverse  axis 
of  an  ellipse  make  with  that  axis,  on  the  same  side. 

Let  #,  y,  be  the  co-ordinates  of 
any  point,  as  P,  and  x',  yr,  the  co 
ordinates  of  the  point  A'. 

A'  I 

Then  the  equation  of   a  line      ' 
which    passes    through    the   two 
points  A'  and  P,  (Prop.  3,  Chap. 
1,)  will  be 


148  ANALYTICAL    GEOMETRY. 

y  —  y'—a(x  —  x').  00 

The  equation  of  the  line  which  passes  through  the 
points  A  and  P,  will  be  of  the  form 

y—  y"=a'(x—  x"}.  (2) 

For  the  given  point  A',  we  have  2//==0,  and  xf=  —  A. 
"Whence  eq.  (1)  becomes 

y=a{x+A).  (3) 

For  the  given  point  A  we  have  #"=0,  and  x"=A,  which 
values  substituted  in  eq.  (2)  give 

y=a'(x—A).  (4) 

As  y  and  x  are  the  co-ordinates  of  the  same  point  P  in 
both  lines,  we  may  combine  eqs.  (3)  and  (4)  in  any  man 
ner  we  please.  Multiplying  them  member  by  member, 
we  have 

yz=aa'(x2—  A2).  (5) 

Because  F  is  a  point  in  the  ellipse,  the  equation  of  the 
curve  gives 

^=J(^-**)=-JV-^).  (6) 

Comparing  eqs.  (5)  and  (6),  we  find 


for  the  equation  sought. 

SCHOLIUM  1.  —  In  case  the  ellipse  becomes  a  circle,  that  is,  in  case 
A=JB,  aa'-f-l=:0,  showing  that  the  angle  A'  PA  would  then  be  a 
right  angle,  as  it  ought  to  be,  by  (Prop.  II,  Chap.  II.) 

7?2 

Because    —  is  less  than  unity,  or  aaf  less  than  1,*  or  radius  ; 

A* 

the  two  angles  PA'  A  and  PAA'  are  together  less  than  90°  ;  there 
fore,  the  angle  at  P  is  obtuse,  or  greater  than  90°. 

SCHOLIUM  2.  —  Since  aa'  has  a  constant  value,  the  sum  of  the  two, 
',  will  be  least  when  a=af. 


*  In  trigonometry  we  learn  that  tan.  x  cot.  £=#2=1.  That  is,  the  pro 
duct  of  two  tangents,  the  sum  of  whose  arcs  is  90°,  is  equal  to  1.  When 
the  sum  is  less  than  90°,  the  product  will  be  a  fraction. 


THE    ELLIPSE.  149 

Hence  the  angle  at  P  will  be  greatest  when  P  is  at  the  vertex 
of  the  minor  axis,  and  the  supplementary  chords  equal ;  and  the 
angle  at  P  will  become  nearer  a  right  angle  as  P  approaches  A  or 
A'. 


PROPOSITION   VI. 

To  find  the  equation  of  a  straight  line  which  shall  be  tangent 
to  an  ellipse. 

Assume  any  two  points,  as 
P'  and  §,  on  the  ellipse,  and 
denote  the  co-ordinates  of  the 
first  by  x1 ',  y1 ',  and  of  the  second 
by  £",  y".  Through  these  points 
draw  a  line,  the  equation  of 
which  (Prop.  4,  Chap.  1,)  is 

y—y'=a(x—x')y  (1) 


in  which 


y— 


x'  —  x" 

"We  must  now  determine  the  value  of  a  when  this  line 
becomes  a  tangent  line  to  the  ellipse. 

Because  the  points  P  and  Q  are  in  the  curve,  the  co 
ordinates  of  those  points  must  satisfy  the  following  equa 
tions  : 


By  subtraction 

Or        A*(y'+y")(y'-y")=—B*(xf+x"}(x'-x"}.    (2) 

Whence  n-jr'^f  —**(*+*) 

x'—x"        A*(y'+y") 

Now  conceive  the  line  to  revolve  on  the  point  P  until 
Q  coincides  with  P,  then  PE  will  be  tangent  to  the  curve. 
But  when  Q  coincides  with  P,  we  shall  have 

y'=y"  and  x'=x". 
13* 


150  ANALYTICAL    GEOMETKY. 

Under  this  supposition,  we  have 


A*yf 
The  value  of  a  put  in  eq.  (1),  gives 


~y 

Reducing          A*yy'+B*xx'=A*y'*+B*tf** 

Or  A*yy'+B*xx'=A*B*. 

This  is  the  equation  sought,  x  and  y  being  the  general 
co-ordinates  of  the  line. 

SCHOLIUM  1.  —  To  find  where  the  tangent  meets  the  axis  of  X, 
we  must  make  y=0. 

This  gives  x=^-=  CT. 

In  case  the  ellipse  becomes  a  circle, 
J3=A,  and  then  the  equation  will  be- 
come  yy'-\-xx'=A2, 

the  equation  for  a  tangent  line  to  a  cir 
cle;  and  to  find  where  this  tangent  meets  the  axis  of  X>  we  make 
y=  0,  and 

x—^—  CT,  as  before. 

In  short,  as  these  results  are  both  independent  of  JB,  the  minor 
axis,  it  follows  that  the  circle  and  all  ellipses  on  the  major  axis  AB 
have  tangents  terminating  at  the  same  point  T  on  the  axis  of  J5T, 
if  drawn  from  the  same  ordinate,  as  shown  in  the  figure. 

SCHOLIUM  2.  —  To  find  the  point  in  which  the  tangent  to  an 
ellipse  meets  the  axis  of  T}  we  make  #=0,  then  the  equation  for 
the  tangent  becomes 

y-—  • 

y    y' 

As  this  equation  is  independent  of  A,  it  shows  that  all  ellipses 
having  the  same  minor  axis,  have  tangents  terminating  in  the  same 
point  on  the  axis  of  Y,  if  drawn  from  the  same  abscissa. 

SCHOLIUM  3.     If  from  CTwe  subtract  CM,  we  shall  have 


THE    ELLIPSE. 


151 


a  common  subtangent  to  a  circle,  and  all  ellipses  which  have  2  A  for 
a  major  diameter.     That  is 


x'  x' 

We  can  also  find  RT  by  the  triangle  PRT,  as  we  have  the  tan 

gent  of  the  angle  at  T,  /—        -)  to  the  radius  1. 
\     A*y'  I 

Whence  we  have  the  following  proportion  : 


The  minus  sign  indicates  that  the  measure  from  T  is  towards  the 
left. 


PROPOSITION   VII. 

To  find  the  equation  of  a  normal  line  to  the  ellipse. 
Since  the  normal  passes  through  the  point  of  tangency, 

its  equation  will  be  in  the  form 

f ff         f\  /^\ 

Because  PN  is  at  right  angles 
to  the  tangent, 

oa'+l=0. 
But  by  the  last  proposition 

a— — - 


Whence  a'=  *JL,  and  this  value  of  a'  put  in  eq.  (1)  gives 

JL>  X 


for  the  equation  sought. 

SCHOLIUM  1.  —  To  find  where  the  normal  cuts  the  axis  of  X,  we 
must  make  ^=0,  then  we  shall  have 


152 


ANALYTICAL    GEOMETRY. 


APPLICATION.  —  Meridians  on  the  earth  are  ellipses;  the  semi- 
major  axis  through  the  equator  is  A=39G3.  miles,  and  the  semi- 
minor  axis  from  the  center  to  the  pole  is  .Z?r=3949.5. 

A  plumb  line  is  everywhere  at  right  angles  to  the  surface,  and 
of  course  its  prolongation  would  be  a  normal  line  like  PJV.  In 
latitude  42°,  what  is  the  deviation  of  a  plumb  line  from  the  center 
of  the  earth  ?  In  other  words,  how  far  from  the  center  of  the 
earth  would  a  plumb  line  meet  the  plane  of  the  equator?  Or,  what 
would  be  the  value  of  CNf 

As  this  ellipse  differs  but  little  from  a  circle,  we  may  take  CR 
for  the  cosine  of  42°,  which  must  be  represented  by  x'.  This  being 
assumed,  we  have 


s'=2945. 


==Cr.Ar     Ans. 


A2     J 

SCHOLIUM  2.  —  To  find  JWR,  the  subnormal,  we  simply  subtract 
(7.^  from  CR,  whence 


A*  A* 

We  can  also  find  the  subnormal  from  the  similar  triangles  PR  T, 
PNR,  thus  : 

TR:RP::RP:RN. 

—A  tf*  :    'r.'  :  —ATR.     Whence  NR= 


PROPOSITION  VIII. 

Lines  drawn  from  the  foci  to  any  point  in  the  ellipse  make 
equal  angles  with  the  tangent  line  drawn  through  the  same  point. 

Let  C  be  the  center  of 
the  ellipse,  PT  the  tangent 
line,  and  PF,  PF',  the 
two  lines  drawn  to  the  foci. 

Denote  the  distance 


—JB*  by  c,  CFf 


THE    ELLIPSE.  153 

by  —  c,  the  angle  FPTby  V,  and  the  tangents  of  the  angles 
P2!Z,  PFT,  by  a  and  a'. 
Now  FPT=PTX—PFT. 

By  trigonometry,  (Eq.  29,  p.  253,  Robinson's  Geometry), 
we  have 

Tan.  ^PT=tan.  (PTX—PFT\ 


That  is,  tan.  F=   "".  (!) 

1-foa 


Prop.  6,  gives  us  a=-^'.  *',?/',  being  the  co-ordi- 

2p 


nates  of  the  point  P. 
Let  x,  y,  be  the  co-ordinates  of  the  point  F,  then  from 

Prop.  4,  Chap.  1,  we  have 

w'  —  y 
a!=¥-  —  *L. 
x'  —  x 

But  at  the  point  _F,  y=Q  and  x—c. 

Whence  af=JL  __ 

xf  —  c 

These  values  of  a  and  af  substituted  in  eq.  (1)  give 
—B*x'__    yf 
A*~yf     x'—c     - 
~ 


A*y'(xf—c}—B*x'yr 
B*cx'—A*      B* 


Tan.  ^  =(At—B*)y?y'—A*cyf  cy'(cx'—  A*)  ctf 
observing  that  J.y2+^^/2=J.2^2,and  J.2—  JB*=c*. 
The  equation  of  the  line  PF  will  become  the  equation  of 
the  line  PF'  by  simply  changing  +c  to  —  c,  for  then  we 
shall  have  the  co-ordinates  of  the  other  focus. 

We  now  have 


tan. 

cy' 
But  if  c  is  made  —  c,  then 

tan.  2PPT=—— 
cy' 


154  ANALYTICAL   GEOMETRY. 

As  these  two  tangents  are  numerically  the  same,  differ 
ing  only  in  signs,  the  lines  are  equally  inclined  to  the 
straight  lines  from  which  the  angles  are  measured,  or  the 
angles  are  supplements  of  each  other. 

Whence  FPT+F'PT=18Q. 

But  FfPH+F'PT=18Q. 

Therefore  FPT=F'PH. 

Cor.  The  normal  being  perpendicular  to  the  tangent, 
it  must  bisect  the  angle  made  by  the  two  lines  drawn 
from  the  tangent  point  to  the  foci. 

SCHOLIUM. — Any  point  in  the  curve  may  be  considered  as  a 
point  in  a  tangent  to  the  curve  at  that  point. 

It  is  found  by  experiment  that  light,  heat  and  sound,  after  they 
approach  to,  are  reflected  off,  from  any  reflecting  surface  at  equal 
angles ;  that  is,  for  any  ray,  the  angle  of  reflection  is  equal  to  the 
angle  of  incidence. 

.  Therefore,  if  a  light  be  placed  at  one  focus  of  an  ellipsoidal  re 
flecting  surface,  such  as  we  may  conceive  to  be  generated  by  revolv 
ing  an  ellipse  about  its  major  axis,  the  reflected  rays  will  be  con 
centrated  at  the  other  focus.  If  the  sides  of  a  room  be  ellipsoidal, 
and  a  stove  is  placed  at  one  focus,  the  heat  will  be  concentrated  at 
the  other. 

Whispering  galleries  are  made  on  this  principle,  and  all  theaters 
and  large  assembly  rooms  should  more  or  less  approximate  to  this 
figure.  The  concentration  of  the  rays  of  heat  from  one  of  these 
points  to  the  other,  is  the  reason  why  they  are  called  the  foci,  or 
burning  points. 


PROPOSITION    IX. 

The  product  of  the  tangents  of  the  angles  that  a  tangent  line 
to  the  ellipse  and  a  diameter  through  the  point  of  contact,  make 
with  the  major  axis  on  the  same  side,  is  equal  to  minus  the 
square  of  the  semi-minor  divided  by  the  square  of  the  semi- 
major  axis. 


THE   ELLIPSE.  155 

Let  PT  be  the   tangent 
line  and  PPf  the  diameter 

through  the  point  of  contact    ///^          ^x^^^x^Xj    X 
B$  and  denote  the  co-ordi 
nates  of  P  by  x',  y'.     The 
equation  of  the  diameter  is 


in  which  af  is  the  tangent  of  the  angle  PCT. 

Since  this  line  passes  through  the  point  P,  we  must 
have 

yf—a'xf 

Whence  a'=^  (1) 

xf 

For  the  tangent  of  the  angle  P  J!Xwe  have 


Multiplying  eqs.  (1)  and  (2),  member  by  member,  we 
find 

«,'--£ 

A* 

SCHOLIUM.  —  The  product  of  the  tangents  of  the  angles  that  a 
diameter  and  a  tangent  line  through  its  vertex  make  with  the  major 
axis  of  an  ellipse  is  the  same  (Prop.  5)  as  that  of  the  tangents  of 
the  angles  that  supplementary  chords  drawn  through  the  vertices  of 
the  major  axis  make  with  it. 

Hence,  if  a=a,  then  a'=a'.  That  is,  if  the  diameter  is  paral 
lel  to  one  of  the  chords,  the  tangent  line  will  be  parallel  to  the  other 
chord,  and  conversely.  This  suggests  an  easy  rule  for  drawing  a 
tangent  line  to  an  ellipse  at  a  given  point,  or  parallel  to  a  given  line. 


OF  THE  ELLIPSE  REFERRED  TO  CONJUGATE  DIAMETERS. 

Two  diameters  of  an  ellipse  are  conjugate  when  either 
is  parallel  to  the  tangent  lines  drawn  through  the  vertices 
of  the  other. 


156  ANALYTICAL    GEOMETEY. 

Since  a  diameter  and  the  tangent  line  through  its  ver 
tex  make,  with  the  major  axis,  angles  whose  tangents 
satisfy  the  equation 


it  follows  that  the  tangents  of  the  angles  which  any  two 
conjugate  diameters  make  with  the  major  axis  must  also 
satisfy  the  same  equation. 

Now  let  m  he  the  angle  whose  tangent  is  a,  and  n  be 
the  angle  whose  tangent  is  a',  then 


cos.  m  cos.  n 


Substituting  these  values  in  the  last  equation,  and  re 
ducing,  we  obtain 

A2  sin.  m  sin.  n-\-J52  cos.  m  cos.  7i=0, 
which  expresses  the  relation  which  must  exist  between  A, 
B,  m,  and  n,  to  fix  the  position  of  any  two  conjugate  di 
ameters  in  respect  to  the  major  axis,  and  this  equation  is 
called  the  equation  of  condition  for  conjugate  diameters. 

In  this  equation  of  condition,  m  and  n  are  undeter 
mined,  showing  that  an  infinite  number  of  conjugate  di 
ameters  might  be  drawn,  but  whenever  any  value  is  as 
signed  to  one  of  these  angles,  that  value  must  be  put  in 
the  equation,  and  then  a  deduction  made  for  the  value  of 
the  other  angle. 

PBOPOSITION    X. 

To  find  the  equation  of  the  ellipse  referred  to  its  center  and 

conjugate  diameters. 

The  equation  of  the  ellipse  referred  to  its  major  and 
minor  axes,  is 


The  formulas  for  changing  rectangular  co-ordinates 


THE    ELLIPSE.  157 

into  oblique,  the  origin  being  the  same,  are  (Prop.  9, 
Chap.  1,) 

x=xf  cos.  m+yf  cos.  n.          y—x1  sin.  m-\-yf  sin.  n. 
Squaring  these,  and  substituting  the  values  of  x2  and 
y2  in  the  equation  of  the  ellipse  above,  we  have 
(  (A2siu2n+B2cos2n)y'2+(A2sm2m+B2voa2m)x'2  }  ^ijp 
\      +2(J.2siii.m  sin.n+.B2cos.m  coa.n)yfxf          ) 

But  if  we  now  assume  the  condition  that  the  new  axes 
shall  be  conjugate  diameters,  then 

J.2sin.  m  sin.  n+  IPcos.  m  cos.  n=0, 
which  reduces  the  preceding  equation  to  (F) 


which  is  the  equation  required.     But  it  can  be  simplified 
as  follows : 

The  equation  refers  to  the  two  di 
ameters  B"B'  and  D"D'  as  co-ordi 
nate  axes.  For  the  point  B'  we 
must  make  ?/;=0,  then 

Xf2=    2  .  A*W     _= 

-A'2.  (P) 


Designating  CBf  by  A',  and  CD'  by  B'. 
For  the  point  Df  we  must  make  x'=Q.     Then 


A1T& 

From  (P)  we  have  (J.2sin.2m+JB2cos.2m)=±i^-. 

xL 

A2£* 
From  (Q)  (J.2sin.2n+^2cos.27i)=-g72-- 

These  values  put  in  (F)  give 

A  2  T>2  A2T)2 

±  ^2+^-xf2=A2B2. 
B'2  y       A'2 

Whence  Any'2+B'2x'2=  A'2Br2. 

14 


158  ANALYTICAL    GEOMETRY. 

We  may  omit  the  accents  to  xf  and  ?/',  as  they  are  gen 
eral  variables,  and  then  we  have 


for  the  equation  of  the  ellipse  referred  to  its  center  and 
conjugate  diameters. 

SCHOLIUM. — In  this  equation,  if  we  assign  any  value  to  x  less 
than  A' ',  there  will  result  two  values  of  y,  numerically  equal,  and 
to  every  assumed  value  of  y  less  than  B1 ',  there  will  result  two 
corresponding  values  of  x,  numerically  equal,  differing  only  in  signs, 
showing  that  the  curve  is  symmetrical  in  respect  to  its  conjugate 
diameters,  and  that  each  diameter  bisects  all  chords  which  are  paral 
lel  to  the  other. 

OBSERVATION. — As  this  equation  is  of  the  same  form  as  that  of 
the  general  equation  referred  to  rectangular  co-ordinates  on  the 
major  and  minor  axis,  we  may  infer  at  once  that  we  can  find  equa 
tions  for  ordinates,  tangent  lines,  etc.,  referred  to  conjugate  diame 
ters,  which  will  be  in  the  same  form  as  those  already  found,  which 
refer  to  the  axes.  But  as  a  general  thing,  it  will  not  do  to  draw 
summary  conclusions. 

PROPOSITION   XI. 

As  the  square  of  any  diameter  of  the  ellipse  is  to  the  square 
of  its  conjugate,  so  is  the  rectangle  of  any  two  segments  of  the 
diameter  to  the  square  of  the  corresponding  ordinate. 

Let  CD  be  represented  by  A',  and         c    1^ 
CE  by  B',  CH  by  x,  and  GH  by  y, 
then  by  the  last  proposition  we  have 


Which  may  be  put  under  the  form  jy 

Bf2(A'*—xz\.  & 


"Whence          A'2  :  Bf*  :  :  (A'2—x2)  :  y\ 
Or          (2A'Y  :  (ZBJ  :  :  (A'+x)(A'—x)  :  y*. 
Now  2  A1  and  2Bf  represent  the  conjugate  diameters 
D'D,  E'E,  and  since  CH  represents  x,  A'+x=D'H,  and 


THE    ELLIPSE.  159 

A'  —  x=HD.    Also  y=GH.     Hence  the  above  propor 
tions  correspond  to 

(D'D)2  :  (E'E)2  :  :  D'HxHD  :  (GH}\ 
SCHOLIUM.  —  As  x  is  no  particular  distance  from  C,  CF  may 
represent  x,  then  LF  will  represent  y,  and  the  proportion  then  be 
comes 


Comparing  the  two  proportions,  we  perceive  that 


D'H-HD  :  D'F-FD  ::  GH*  :  LF*. 

That  is,  The  rectangle  of  the  abscissas  are  to  one  another  as  the 
squares  of  the  corresponding  ordinates. 

The  same"  property  as  was  demonstrated  in  respect  to  rectangular 
co-ordinates  in  Prop.  3. 

In  the  same  manner  we  may  prove  that 

Eh-hE'  :  Ef-fE'  ::  (hg)*  : 


PBOPOSITION    XII. 

To  find  the  equation  of  a  tangent  line  to  an  ellipse  referred 
to  its  conjugate  diameters. 

Conceive  a  line  to  cut  the  curve  in  two  points,  whose 
co-ordinates  are  x*  ',  ?/',  and  x",  y",  x  and  y  being  the  co 
ordinates  of  any  point  on  the  line. 

The  equation  of  a  line  passing  through  two  points  is 
of  the  form 

y—yf=a(x—xr),  (i) 

an  equation  in  which  a  is  to  be  determined  when  the  line 
touches  the  curve. 

From  the  equation  of  the  ellipse  referred  to  its  conju 
gate  axes  we  have 

A'2yf2+£'2x/2=A'2J3f2. 


Subtracting  one  of  these  equations  from  the  other,  and 
operating  as  in  Prop.  6,  we  shall  find 

Bf2x' 
a=  —  _  . 

A'2y' 


160  ANALYTICAL    GEOMETRY. 

This  value  of  a  put  in  eq.  (1)  will  give 

T?/2™/ 

y—;u'=—        X  (X—X'\ 

2^r      ' 

Reducing,  and    A'2y'y+B/2x'x=A'2B'2, 
which  is  the  equation  sought,  and  it  is  in  the  same  form 
as  that  in  Prop.  6,  agreeably  to  the  observation  made  at 
the  close  of  Prop.  10, 


PROPOSITION    XIII. 

To  transform  the  equation  of  the  ellipse  in  reference  to  con 
jugate  diameters  to  its  equation  in  reference  to  the  axes. 

The  equation  of  the  ellipse  in  reference  to  its  conju 
gate  diameter  is 

A'2y'2+B'2x'2=A'2B'2.  (l) 

And  the  formulas  for  passing  from  oblique  to  rectangu 
lar  axes  are  (Prop.  10,  Chap.  1,) 

,_£sin.  n  —  ycos.n  ,_j/cos.ra  —  xsm.m 

JU   ~™  --  .  tJ  ~~~  -  ^ 


. 

sin.  (ft  —  m)  sin.  (n  —  m) 

These  values  of  xf  and  yf  substituted  in  eq.  (1)  give 


} 
j 


—  2(  A'2  sin.  m  cos.  m+Br2  sin.  n  cos.  n)xy 

Af2JS'2sm.2(n—  m). 

This  equation  must  be  true  for  any  point  in  the  curve, 
x  being  measured  on  the  major  axis,  and  y  the  corres 
ponding  ordinate  at  right  angles  to  it. 

This  being  the  case,  such  values  of  Af,  Bf,  m,  and  n, 
must  be  taken  as  will  reduce  the  preceding  equation  to 
the  well  known  form 


Therefore  we  must  assume 

A'2  cos.2  m+B'*  cos.2  n=  A2.          (1) 
A'2  sin.2  m+Bf*  sin.2  n=B*.  (2) 

=0.  (3) 
(4) 


THE    ELLIPSE. 


161 


The  values  of  m  and  n  must  "be  taken  so  as  to  respond 
to  the  following  equation,  because  the  axes  are  in  fact 
conjugate  diameters. 

^2sin.msin.7i-f--52cos.mcos.7i==0.       (5) 

These  equations  unfold  two  very  interesting  properties. 
SCHOLIUM  1.  —  By  adding  eqs.  (1)  and  (2)  we  find 


Or  44'2+4£/»=< 

That  is,  the  sum  of  the  squares  of  any  two  conjugate  diameters  is 
equal  to  the  sum  of  the  squares  of  the  axes. 

SCHOLIUM  2. — Equation  eq.  (3)  or  (5)  will  give  us  m  when  n  is 
given  -,  or  give  us  n  when  m  is  given. 

SCHOLIUM  3. — The  square  root  of  eq.  (4)  gives 

which  shows  the  equality  of  two  surfaces}  one  of  which  is  obviously 
the  rectangle  of  the  two  axes. 

Let  us  examine  the  other. 

Let  n  represent  the  angle  NCB,  >^M 

and  m  the  angle  PCB.  Then  the 
angle  NCP  will  be  represented 
by  (n — m). 

Since  the  angle  MNK  is  the 
supplement  of  NCP,  the  two  an 
gles  have  the  same  sine  and 


In  the  right-angled  triangle  NKM,  we  have 
1  :  A'  :  :  sin.(n  —  m)  :  MK. 

MK=A'sm.(n—m). 
But  NC=B'. 

Whence 

MK-NC=A'B'sm.(n—  m)—  the  parallelogram  NCPM. 
Four  times  this  parallelogram  is  the  parallelogram  ML,  and  fonr 
times  the  parallelogram  DOB  II,  which  is  measured  by  Ay^B,  is 
equal  to  the  parallelogram  HF.     Hence  eq.  (4)  reveals  this  general 
truth  : 

The  rectangle  which  is  formed  by  drawing  tangent  lines  through 
14*  L 


162  ANALYTICAL    GEOMETRY. 

the  vertices  of  the  axes  of  an  ellipse  is  equivalent  to  any  parallelo 
gram  which  can  be  formed  by  drawing  tangents  through  the  vertices 
of  conjugate  diameters. 

NOTE.  —  The  student  had  better  test  his  knowledge  in  respect  to  the 
truths  embraced  in  scholiums  1  and  3,  by  an  example  : 

Suppose  the  semi-major  axis  of  an  ellipse  is  10,  and  the  semi-minor 
axis  6,  and  the  inclination  of  one  of  the  conjugate  diameters  to  the  axis 
of  X  is  taken  at  30°  and  designated  ~by  m. 

We  are  required  to  find  A'2  and  .Z?'2,  which  together  should  equal 
AZ+BZ,  or  136,  and  the  area  NCPM,  which  should  equal  AB,  or  60, 
if  the  foregoing  theory  is  true. 

Equation  (5)  will  give  us  the  value  of  n  as  follows  : 

100-  Jtan./H-36-i-N/3=0. 

n  36v/3 

Or  tan.Tinr  —  _  L— 

100  ' 

Log.  36  +i  log.  3  —  log.  100  plus  10  added  to  the  index  to  corres 
pond  with  the  tables,  gives  9.794863  for  the  log.  tangent  of  the  angle  n, 
which  gives  31°  56'  42",  and  the  sign  being  negative,  shows  that  31° 
66'  42"  must  be  taken  below  the  axis  of  JT,  or  we  must  take  the  sup 
plement  of  it,  NCB,  for  n,  whence 

71=148°  3'  18",  and  (n—  m)=118°  3'  18". 
To  find  A'*  and  .#'«,  we  take  the  formulas  from  Prop.  10. 

100-36      =3600=69  2g 


52 

3600 

99+25-92 
66-77.     And  their  sum=136. 
This  agrees  with  scholium  1. 

As  radius  10.000000 

Is  to  4'J(log.69.23)  0.920147 

So  is  sine       (n—  m)  61°  56'  42"  9.945713 


log.  MK=         0.865860 
Log.  B'=.\  log.  (66.77) 0.912290 

60.  log.  60=    1.778150 


THE    ELLIPSE. 


163 


PROPOSITION   XIY. 

To  find  the  general  polar  equation  of  an  ellipse. 

If  we  designate  the  co-ordinates 
of  the  pole  P,  by  a  and  6,  and  es 
timate  the  angles  v  from  the  line 
PX'  parallel  to  the  transverse  axis, 
we  shall  have  the  following  formu 
las  : 

x=a+r  cos.v.    y=b+r  sin  v. 

These  values  of  x  and  y   substituted  in  the  general 
equation  A2y2+J32x2=A2lF, 

will  produce 
A2  sin.2*; 


for  the  general  polar  equation  of  the  ellipse. 

SCHOLIUM  1. — When  P  is  at  the  center,  a— 0,  and  b=Q,  and 
then  the  general  polar  equation  reduces  to 


a  result  corresponding  to  equations  (P)  and  (  Q)  in  Prop.  10. 

SCHOLIUM   2. — When   P  is  on  the   curve  J.252- 
therefore 

"•-•-"-  sin.i> 


This  equation  will  give  two  values  of  r,  one  of  which  is  0,  as  it 
should  be.  The  other  value  will  correspond  to  a  chord,  according 
to  the  values  assigned  to  a,  b,  and  v.  Dividing  the  last  equation 
by  the  equation  r=0,  and  we  have 

sin.v      A 


The  value  of  r  in  this  equation  is  the  value  of  a  chord. 
When  the  chord  becomes  0,  the  value  of  r  in  the  last  equation 
becomes  0  also,  and  then 


164  ANALYTICAL    GEOMETRY. 


Or 


a  result  corresponding  to  Prop.  6,  as  it  ought  to  do,  because  the 
radius  vector  then  becomes  tangent  to  the  curve. 

SCHOLIUM  3. — When  P  is  placed  at  the  extremity  of  the  major 
axis  on  the  right,  and  if  vr^O,  then  sin.  vmO,  and  cos.  #=1  a=Af 
and  6  — 0 ;  these  values  substituted  in  the  general  equation  will  re 
duce  it  to  J2V-f2.Z?a^4r=0, 
which  gives  r=Q,  and  r=—  2A,  obviously  true  results. 

When  P  is  placed  at  either  focus,  then  a=*/A* — J3*=c,  and 
5=0.     These  values  substituted,  and  we  shall  have 

It  is  difficult  to  deduce  the   values    of  r  from    this    equation, 
therefore  we  adopt  a  more  simple  method. 

Let  F  be  the  focus,  and  FP  any  radi 
us,  and  put  the  angle  PFD=v. 

By  Prop.  1,  of  the  ellipse,  we  learn 
that 

(!) 

an  equation  in  which  c—  -s/J.8 — £*,  and  x  any  variably  distance 
CD. 

Take  the  triangle  PD F,  and  by  trigonometry  we  have 
1  :  r ::  cos.v  :  c-\-x. 

Whence  x=:r  cos.v — c. 

This  value  of  x  placed  in  (1),  will  give 

cr.  cos.v— ca 
r=A+ J- 

Whence  (A — c  cos.v}r=A* — ca 

A9— c9 


Or 


A  —  c  cos.v 

This  equation  will  correspond  to  all  points  in  the  curve  by  giving 
to  cos.v  all  possible  values  from  1  to  —  1.  Hence,  the  greatest 
value  of  r  is  (  J.-j-c),  and  the  least  value  (J.  —  c),  obvious  results 
when  the  polar  point  is  at  F. 


THE    ELLIPSE.  165 

The  above  equation  may  be  simplified  a  little  by  introducing  the 
ecci'iitririty.  The  eccentricity  of  an  ellipse  is  the  distance  from  the 
center  to  either  focus,  when  the  semi-major  axis  is  taken  as  unity. 
Designate  the  eccentricity  by  e,  then 

1  :  e=  A  :  c. 
Whence  c=eA. 

Substituting  this  value  of  c  in  the  preceding  equation,  we  have 

ea) 


A  —  eA  cos.  v      1  —  e  cos.  v 
This  equation  is  much  used  in  astronomy. 

PROPOSITION   XV.—  PROBLEM. 

Given  the  relative  values  of  three  different  radii,  drawn  from 
the  focus  of  an  ellipse,  together  with  the  angles  between  them, 
to  find  the  relative  major  axis  of  the  ellipse,  the  eccentricity, 
ami  the  position  of  the  major  axis,  or  its  angle  from  one  of  the 
given  radii. 

Let  r,  r',  and  r",  represent  the 
three  given  radii,  m  the  angle  be 
tween  r  and  r',  and  n  that  between 
r  and  r".  The  angle  between  the 
radius  r  and  the  major  axis  is  sup 
posed  to  be  unknown,  and  we  therefore,  call  it  x. 

From  the  last  proposition,  we  have 


1  —  e  cos.  x 


r= 


1  —  e  cos.  (x+m) 

A(l—e2) 
/'=  _  ^  _  L  _        (3\ 

1  —  e  cos.  (x+rt) 

Equating  the  value  of  A(l  —  e2)  obtained  from  eqs.  (1) 
and  (2),  and  we  have 

r  —  re  cos.  x—rf  —  rfe  cos.  (x+m) 


166  ANALYTICAL    GEOMETRY. 

r  _  rt 

~r  cos.  x  —  r'  cos.  (x+m). 
In  like  manner  from  eqs.  (1)  and  (3),  we  have 
r  —  re  cos.  x=r"  —  r"e  cos.  (x+ri). 
_  r—r"  __ 

~~  r  cos.  x  —  r"  cos.  (x+ri) 

Equating  the  second  members  of  eqs.  (4)  and  (5),  we 
have 

r—  r>  _____  r—r"  _ 

r  cos.x  —  r1  cos.(x+m)"~r  cos.x  —  r"  cos.(x+ri) 

Whence,        r~  r'  =r  coa.  x—r'  eoa.  (x+m) 


r  —  r"     r  COs.  x  —  r" 
r  cos.  x  —  r'  cos.  x  cos.  m+r'  sin.  x  sin.  m 
~r  cos.  x  —  r"  cos.  x  cos.  n+r"  sin.  x  sin.  n 

r  —  r1  cos.  m+r'  sin.  m  tan.  x 
~~r  —  r"  cos.  n+  r"  sin.  n  tan.  x 
For  the  sake  of  brevity,  put  r  —  rf=d, 

r  —  r"=d',  the  known  quantity  r  —  rf  cos.  m—a, 
and  r  —  r"cos.w=  b.     Then  the  preceding  equation  becomes 
d    a+r'sin.m  tan.z 
d'~b+r"sm.n  tan.z' 
From  which  we  get  successively 

db+dr"  sin.  n  tan.  x=adf+dfrl  sin.  m  tan.  # 
(dr"  sin.  TI  —  e^r'  sin.  m)  tan.  x—ad'  —  db, 

ad'  —  db 

tan.  x=-j~rf  —  -.  -  17-7  —  :  --  > 
dr  sin.  n  —  d  r'  sm.m 

The  value  of  x  from  this  equation  determines  the  posi 
tion  of  the  major  axis  with  respect  to  that  of  r,  which  is 
supposed  to  be  known,  as  it  may  be  by  observation.  * 

Having  x,  eq.  (4)  or  (5)  will  give  e  the  eccentricity.  If 
the  values  of  e  found  from  these  equations  do  not  agree, 
the  discrepancy  is  due  to  errors  of  observation,  and  in 
such  cases  the  mean  result  is  taken  for  the  eccentricity. 


THE    ELLIPSE.  167 

Equations  (1),  (2)  jind  (3)  contain  A,  the  semi-major 
axis,  as  a  common  factor  in  their  second  members.  This 
factor,  therefore,  does  not  affect  the  relative  values  of  r, 
r'  and  r",  and  as  it  disappears  in  the  subsequent  part  of 
the  investigation,  it  shows  that  the  angle  x  and  the  eccen 
tricity  are  entirely  independent  of  the  magnitude  of  the 
ellipse.  To  apply  the  preceding  formulas,  we  propose 
the  following 

EXAMPLE. 

On  the  first  day  of  August,  1846,  an  astronomer  observed 
the  sun's  longitude  to  be  128°  47'  31",  and  by  comparing  this 
observation  with  observations  made  on  the  previous  and  subse 
quent  days,  he  found  its  motion  in  longitude  was  then  at  the 
rate  of  57'  24".  9  per  day.  By  like  observations  made  on  the 
first  of  September,  he  determined  the  sun's  longitude  to  be  158° 
37'  46",  and  its  mean  daily  motion  for  that  time  58'  6"  6  ;  and 
at  a  third  time,  on  the  Wth  of  October,  the  observed  longitude 
was  196°  48'  4",  and  mean  daily  motion  59'  22".  9.  From 
these  data  are  required  the  longitude  of  the  solar  apogee,  and  the 
eccentricity  of  the  apparent  solar  orbit. 

It  is  demonstrated  in  astronomy  that  the  relative  dis 
tances  to  the  sun,  when  the  earth  is  in  different  parts  of 
its  orbit,  must  be  to  each  other  inversely  as  the  square  root 
of  the  sun's  apparent  angular  motion  at  the  several  points  ; 
therefore,  (r)2,  (r')2,  and  (r")2,  must  be  in  the  proportion  of 

J_          J_  .,  and  _  J_ 

57'  24"  9      58'  6"  6  59'  22"  9 

Or  as  the  numbers 

J_         J_  ,  and    _!_ 
3444.9'     3486.6  3562.9 

Multiply  by  3562.9  and  the  proportion  will  not  be 
changed,  and  we  may  put 


/3562.9U        r,     /3562.9U 
\  3444.97  '  V  3486.6  /  ' 


168  ANALYTICAL    GEOMETRY. 

By  the  aid  of  logarithms  we  soon  find 

r=1.016982  r'=1.010857  and  /'=!. 

Hence    r—rf=d=  0.006125,     r—r"=d'= 0.016982. 
158°  37'  46"  196°  48'    4" 

128    47    31  128    47  31 


m=  29    50    15       71=  68     0    33 
To  substitute  in  our  formulas,  we  must  have  the  natu 
ral  sine  and  cosine  of  m  and  n. 

sin.  m=sin.  29°  50'15"=  0.497542,  cos.=  0.867440. 
sin.  n=sin.  68°  0'  33"=0.927238,  cos.  =0.374472. 
r—  rr  cos.ra=a=0.140124. 
r  —  r"  cos.  71=6=0.642510. 
0^=0.0023695,     ^6=0.00393537. 
d'r'  sin.  m=0.008538616, 
dr"  sin.  71=0.005679332. 
These  values  substituted  in  the  formula 
x_  ad'  —  db  db  —  adr 


-  _^____^_ 

.7i  —  ^V'sin.m    ^V'sin.m  —  dr"  sin.w' 
give 

tan  y==-QQ156586==15.6586 
.00285928     28.5928 

Log.  15.6586  plus  10  to  the  index=  11.194746 
Log.  28.5928  1.456224 

Log.    tan.    28°  42'  45"  9.738522 

Long,  of  r  128°  47'  31" 

Long,  apogee  100°    4'  46" 

According  to  observation,  the  longitude  of  the  solar 
apogee  on  the  1st  of  January,  1800,  was  99°  30r  8"39, 
and  it  increases  at  the  rate  of  61"9  per  annum.  This 
would  give,  for  the  longitude  of  the  apogee  on  the  1st  of 
January,  1861,  100°  33'  03"54. 

To  find  e,  the  eccentricity,  we  employ  eq.  (5),  which  is 


THE     PARABOLA.  169 

r—r" 

g=        __  . 

r  cos.z  —  r"  cos.(x+ri) 

Whence,  by  substituting  the  values  of  r,  r",  cos.  x,  etc., 
we  find 

0.016982  .016982 


= 
~ 


r  cos.  28°42/45'/—  cos.  9604318"     .891891-f  .11694 


1.0088 

CHAPTER  IV. 
THE  PARABOLA. 

To  describe  a  parabola.' 

Let  CD  be  the  directrix,  and  F  the 
focus.  Take  a  square,  as'D-B(7,  and 
to  one  side  of  it,  GB,  attach  a  thread, 
and  let  the  thread  be  of  the  same 
length  as  the  side  GB  of  the  square. 
Fasten  one  end  of  the  thread  at  the  point  6r,  the  other 
end  at  F. 

Put  the  other  side  of  the  square  against  CD,  and  with 

a  pencil,  P,  in  the  thread,  bring  the  thread  up  to  the  side 

of  the  square.     Slide  one  end  of  the  square  along  the 

line   CD,  and  at  the  same  time  keep  the  thread  close 

against  the  other  side,  permitting  the  thread  to  slide 

round  the  pencil  P.     As  the  side  of  the  square,  _RD,  is 

moved  along  the  line  CD,  the  pencil  will  describe  the 

curve  represented  as  passing  through  the  points  V  and  P. 

GP+PF=  the  thread. 

GP+P£=  the  thread. 

By  subtraction  PF—P£=0,  or  PF=PB. 

This  result  is  true  at  any  and  every  position  of  the 
point  P  ;  that  is,  it  is  true  for  every  point  on  the  curve. 

Hence,  FV=VH. 

15 


170  ANALYTICAL    GEOMETRY. 

If  the  square  be  turned  over  and  moved  in  the  oppo 
site  direction,  the  other  part  of  the  parabola,  on  the  other 
side  of  the  line  FH  may  be  described. 


PROPOSITION   I. 

To  find  the  equation  of  the  parabola. 

Take  the  axis  of  the  parabola  for 
the  axis  of  abscissas  and  the  line  at 
right  angles  to  it  through  the  vertex 
for  the  axis  of  ordinates. 

The  perpendicular  distance  from  the      "H  V .  F  D 
focus  F  to  the  directrix  BH,  is  called  \ 

p,  a  constant  quantity,  and  when  this  constant  is  large, 
we  have  a  parabola  on  a  large  scale,  and  when  small,  we 
have  a  parabola  on  a  small  scale. 

By  the  definition  of  the  curve,  V  is  midway  between  F 
and  the  line  BH,  and  PF=PB. 

Put  VD=x  and  PD=y,  and  operate  on  the  right  an 
gled  triangle  PDF. 


(FD)2+(PD)2=(PF)2. 
That  is,  (XT- %p)*+f=(x+±p)2. 

Whence        y2=2px,  the  equation  sought. 

Cor.  1.  If  we  make  cc=0,  we  have  y=Q  at  the  same  time, 
showing  that  the  curve  passes  through  the  point  "F,  cor 
responding  to  the  definition  of  the  curve. 

As  ?/=±v/^j9x,  it  follows  that  for  every  value  of  x 
there  are  two  values  of  y,  numerically  equal,  one  -f ,  the 
other  — ,  which  shows  that  the  curve  is  symmetrical  in 
respect  to  the  axis  of  X. 

Cor.  2.  If  we  convert  the  equation  y2=2px  into  a  pro 
portion,  we  shall  have 

x  :  y  :  :  y  :  2p, 


THE    PARABOLA.  171 

a  proportion  showing  that  the  parameter  of  the  axis  is  a 
third  proportional  to  any  abscissa  and  its  corresponding  ordi- 
nate. 

Cor.  3.  If  we  substitute  \p  for  x  in  the  equation  y2—2px 
we  get 

y=p  or  2y=2p. 

That  is  the  parameter  of  the  axis  of  the  parabola  is  equal 
to  the  double  ordinate  through  the  focus,  or,  it  is  equal  to  four 
times  the  distance  from  the  vertex  to  the  directrix. 

PROPOSITION   II. 

The  squares  of  ordinates  to  the  axis  of  the  parabola  are  to 
one  another  as  their  corresponding  abscissas. 

Let  x,  y,  be  the  co-ordinates  of  any  point  P,  and 
the  co-ordinates  of  any  other  point  in  the  curve. 
Then  by  the  equation  of  the  curve  we  must  have 
y*=2px.  (1) 

y'2=2px',  (2) 

By  division  ^7iM — T 

Whence  y2 :  y'2  :  :  x  :  d . 

PROPOSITION    III. 

To  find  the  equation  of  a  tangent  line  to  the  parabola. 

Draw  the  line  SPQ  intersecting 
the  parabola  in  the  two  points  P  and 
Q.  Denote  the  co-ordinates  of  the 
first  point  by  a/,  y' ,  and  of  the  sec 
ond,  by  x",  y". 

The  equation  of  the  straight  line  T~"~ 
passing  through  these  points  is 

(1) 


172  ANALYTICAL   GEOMETEY. 

y'  -y" 

in  which  a  is  equal  to  x,     xn 

It  is  now  required  to  find  the  value  of  a  when  the 
point  Q  unites  with  P,  or,  when  the  secant  line  "becomes 
a  tangent  line  at  the  point  P. 

Since  P  and  Q  are  on  the  parabola  we  must  have 
y'2=2pxf 

And  y"2=2px" 

Whence  y'  *—y"*=  2p(x'  —x") 

or  (y-yw  +/)=%>(*'-*') 

y'—u"      2p—X 

Therefore          a=  V-^/=  -TIT/ 
x  —  x'   y  +y 

Substituting  this  value  of  a  in  eq.  (1)  we  have  for  the 
equation  of  the  secant  line. 


Now  if  this  line  he  turned  about  P  until  Q  coincides 
with  P  we  shall  have  y"=yf  and  the  line  becomes  tangent 
to  the  curve  at  the  point  P. 

73 

Under  this  supposition  the  value  of  a  becomes  j  and 
equation  (2)  reduces  to 


Or  y  y  —  y'  *  ^ 

But  y'2—2px'',  substituting  this  value  y12  in  the  last 
equation,  transposing  and  reducing,  we  have  finally 

yy'=p(x+x')  (3) 

for  the  equation  of  the  tangent  line. 

Cor.  To  find  the  point  in  which 
the  tangent  meets  the  axis  of  JT, 
we  must  make  y=0,  this  makes 


Or  x'= — x. 


THE    PARABOLA.  173 

That  is,  VD=  VT,  or  the  sub-tangent  is  bisected  by  the 
vertex. 

Hence,  to  draw  a  tangent  line  from  any  given  point,  as 
P,  we  draw  the  ordinate  PD,  then  make  TV=  VD,  and 
from  the  point  T  draw  the  line  2P,  and  it  will  be  tan 
gent  at  P,  as  required. 


PROPOSITION    IV. 

To  find  the  equation  of  a  normal  line  in  the  parabola. 

The  equation  of  a  straight  line  passing  through  the 
point  P  is 

y-y'=a(x-x'}.  (1)  . 

Let  #!,  3/j,  be  the  general  co-ordinates  of  another  line 
passing  through  the  same  point,  and  a'  the  tangent  of 
the  angle  it  makes  with  the  axis  of  the  parabola,  its 
equation  will  then  be 

yl—y'=af(xl—xr}.  (2) 

But  if  these  two  lines  are  perpendicular  to  each  other, 
we  must  have 

aaf=—  1.  (3) 

But  since  the  first  line  is  a  tangent, 


This  value  substituted  in  eq.  (3)  gives 

il' 

a'=—^-. 
P 

And  this  value  put  in  eq,  (2)  will  give 


for  the  equation  required. 
15* 


174  ANALYTICAL    GEOMETRY. 

Cor.   1.     To   find  the   point  in 

which  the  normal  meets  the  axis  of 

X,  we  must  make  y ,  =  0.     Then  by 

a  little  reduction  we  shall  have 

p=xl—  xf. 

But  VC=xl9  and  VD=xf.     Therefore  DC=p,  that  is, 
The  sub-normal  is  a  constant  quantity,  double  the  distance 

between  the  vertex  and  focus. 

Cor.  2.  Since  TV=  VD,  and  VF=$DC,  TF=FC. 
Therefore,  if  the  point  F  be  the  center  of  a  circle  of 
which  the  radius  is  FC,  the  circumference  of  that  circle 
will  pass  through  the  point  P,  because  TPC  is  a  right 
angle.  Hence  the  triangle  PFTis  isosceles.  Therefore, 
If  from  the  point  of  contact  of  a  tangent  line  to  the  parabola 
a  line  be  drawn  to  the  focus  it  will  make  an  angle  with  the  tan 
gent  equal  to  that  made  by  the  tangent  with  the  axis. 

Cor.  3.  Now  as  V  bisects  TD  and  VB  is,  parallel  to 
PZ>,  the  point  B  bisects  TP.  Draw  FB,  and  that  line 
bisects  the  base  of  an  isosceles  triangle,  it  is  therefore 
perpendicular  to  the  base.  Hence,  we  have  this  general 
truth : 

If  from  the  focus  of  a  parabola  a  perpendicular  be  drawn  to 
any  tangent  to  the  curve,  it  will  meet  the  tangent  on.  the  axis  of  Y. 

Also,  from  the  two  similar  right-angled  triangles,  FB  V 
and  FB  T,  we  have 

TF-.FB::  FB  :  FV. 

Whence  ~BF*=  TF  -  FV. 

But  FV  is  constant,  therefore  (ft}?}2  varies  as  TF,  or  as  its 
equal  PF. 

SCHOLIUM. — Conceive  a  line  drawn  par 
allel  to  the  axis  of  the  parabola  to  meet 
the  curve  at  P;  that  line  will  make  an 
angle  with  the  tangent  equal  to  the  angle 
FTP.  But  the  angle  FTP  is  equal  to 
the  angle  FPT;  hence  the  L  LPA=ih& 


THE     PARABOLA.  175 

[_  FPT.  Now,  since  light  is  incident  upon  and  reflected  from  sur 
faces  under  equal  angles,  if  we  suppose  LP  to  be  a  ray  of  light  in 
cident  at  P,  the  reflected  ray  will  pass  through  the  focus  F,  and 
this  will  be  true  for  rays  incident  on  every  point  in  the  curve; 
hence,  if  a  reflecting  mirror  have  a  parabolic  surface,  all  the  rays  of 
light  that  meet  it  parallel  with  the  axis  will  be  reflected  to  the  focus  ; 
and  for  this  reason  many  attempts  have  been  made  to  form  perfect 
parabolic  mirrors  for  reflecting  telescopes. 

If  a  light  be  placed  at  the  focus  of  such  a  mirror,  it  will  reflect 
all  its  rays  in  one  direction  j  hence,  in  certain  situations,  parabolic 
mirrors  have  been  made  for  lighthouses  for  the  purpose  of  throwing 
all  the  light  seaward. 

PROPOSITION    Y. 

If  two  tangents  be  drawn  to  a  parabola  at  the  extremities 
of  any  chord  passing  through  the  focus,  these  tangents  will  be 
perpendicidar  to  each  other,  and  their  point  of  intersection  will 
be  on  the  directrix. 

Let  PPf  be  any  chord  through  the  focus 
of  the  parabola,  and  P  T,  Pr  T  the  tangents 
drawn  through  its  extremities.  Through 
T,  their  intersection,  draw  BBf  perpendic 
ular  to  the  axis  HF,  and  from  the  focus  let 
fall  the  perpendiculars  Ft,  Ft1  upon  the 
tangents  producing  them  to  intersect  BB' 
at  B  and  B'  .  Draw,  also,  the  lines  PB,  P'B',  and  it'. 

First.  —  The  equation  of  the  chord  is 

(1) 


and  of  the  parabola 

f=2px  (2) 

Combining  eqs.  (1)  and  (2)  and  eliminating  x9  we  find 
that  the  ordinates  of  the  extremities  of  the  chord  are  the 
roots  of  the  equation 


176  ANALYTICAL    GEOMETBY. 

"Whence  _ 

y,_P+pS*+J  and  ,f=P-P^+i 

a  a 

Therefore  the  tangents  of  the  angles  that  the  tangent 
lines  at  the  extremities  of  the  chord  make  with  the  axis 
are 

P  a 


The  product  of  these  tangents  is 
a  a 


_ 
1—     tf+~ 

Whence  we  conclude  that  the  tangent  lines  are  perpen 
dicular  to  each  other. 

Second.  —  Because  the  AtFtf  is  right-angled  and  FV  is 
a  perpendicular  let  fall  from  the  vertex  of  the  right  angle 
upon  the  hypothenuse,  we  have  (Th.  25,  B.  II,  Geom.) 

Ff  :  Ft'*  :  :  Vt  :  Vt' 

and  because  W  and  BBf  are  parallel,  (Cor.  3,  Prop.  4),  we 
also  have 

Ff  '.Fi'*i:FBi:  FB>* 
nHB:  HB' 
But  (Cor.  3,  Prop.  4,) 

J?  :  Ft'*  :  :  FP  :  FP' 
Therefore 

FP  :  FPf  ::HB:  HB' 

Hence  the  lines  PB,  P'Bf  are  parallel  to  the  axis  of 
the  parabola,  and  (Cor.  2,  Prop.  4,)  the  angles  BPt  and 
tPF  are  pqual.  Therefore  the  right-angled  triangles  BPt 
and  tPF  are  equal,  and  PB=PF.  In  the  same  way  we 
prove  that  P'B'=PfF.  The  line  BBf  is  therefore  the 
directrix  of  the  parabola. 

Cor.  Conversely:  If  two  tangents  to  the  parabola  are  per 
pendicular  to  each  other,  the  chord  joining  the  points  of  contact 
passes  through  the  focus. 


THE    PARABOLA.  177 

For,  if  not,  draw  a  chord  from  one  of  the  points  of 
contact  through  the  focus,  and  at  the  extremity  of  this 
chord  draw  a  third  tangent.  Then  the  second  and  third 
tangents  being  both  perpendicular  to  the  first,  must  be 
parallel. 

But  a  tangent  line  to  a  parabola,  at  a  point  whose  or- 

dinate  is  ?/',  makes  with  the  axis  an  angle  having  ^    for 

& 

its  tangent ;  and  as  no  two  ordinates  of  the  parabola  are 
algebraically  equal,  it  is  impossible  that  the  curve  should 
have  parallel  tangent  lines. 


PROPOSITION  VI. 

To  find  the  equation  of  the  parabola  referred  to  a  tangent  line 
and  the  diameter  passing  through  the  point  of  contact  as  the 
co-ordinate  axes. 

Let  Vbe  the  vertex  and  "PLI'the 
axis   of   the  parabola.     Through  'Y 

any  point  of  the  curve,  as  P,  draw 
the  tangent  PFand  the  diameter 
PR,  and  take  these  lines  for  a  sys 
tem  of  oblique  co-ordinate  axes. 
From  a  point  M,  assumed  at  plea 
sure,  on  the  parabola,  draw  MR 
parallel  to  P  Y  and  MS  perpendicular  to  VX  ;  also,  draw 
PQ  perpendicular  to  VX. 

Let  our  notation  be  VQ=c,  PQ=b,  VS'=x,  MS'=y, 
PR=x',  MR=yf  and  [__MRS=[_MIl'S'=m;  then  the 
formulas  for  changing  the  reference  of  points  from  a  sys 
tem  of  rectangular  to  a  system  of  oblique  co-ordinate 
axes  having  a  different  origin,  give,  by  making  [_n=0, 
x=  c-\-  x'  +  ?/rcos.w 


M 


178  ANALYTICAL    GEOMETRY. 

These  values  of  x  and  y  substituted  in  the  equation  of 
the  parabola  referred  to  V  as  the  origin  which  is 

y*=2px  (1) 

will  give 
b2+2by'8iu.m+y'2sm.2m—<2ipc+2pxf+2pyfcos.m    (2) 

Because  P  is  on  the  curve,  65=2pc,  and  because  JRM 
is  parallel  to  the  tangent  P  Y,  we  also  have  (Prop.  3,) 


cos.m     b 

Whence  26?/'sin.m=2p/  cos.ra 

By  means  of  these  relations  we  can  reduce  eq.  (2)  to 


Or 

If  we  denote  --  by  2p'  the  equation  of  the  curve 
sm.2m    J 

referred  to  the  origin  Pand  the  oblique  axes  PJT,  PY, 
becomes 

y"=»2pV 

an  equation  of  the  same  form  as  that  before  found  when 
the  vertex  V  was  the  origin  and  the  axes  rectangular. 

Cor.  1.  Since  the  equation  gives  y'=±^2p'xf,  that  is 
for  every  value  of  x'  two  values  of  y1  ,  numerically  equal, 
it  follows  that  every  diameter  of  the  parabola  bisects  all  chords 
of  the  curve  drawn  parallel  to  a  tangent  through  the  vertex  of 
the  diameter. 

Cor.  2.  The  squares  of  the  ordinates  to  any  diameter  of 
the  parabola  are  to  each  other  as  their  corresponding  abscissas. 

Let  x,  y  and  x',  y'  be  the  co-ordinates  of  any  two 
points  in  the  curve,  then 


Whence  IL=5_ 

yfZ          Xf 


THE   PARABOLA. 


179 


Or  y*  :  yf2  :  :  x  :  xf 

Cor.  3.  By  a  process  in  no  respect  differing  from  that 
followed  in  proposition  3  we  shall  find 


for  the  equation  of  a  tangent  line  to  the  parabola  when 
referred  to  any  diameter  and  the  tangent  drawn  through 
its  vertex  as  the  co-ordinates  axes. 

If,  in  this  equation,  we  make  y=0  we  get 
x+x'=Q  or  x= — x1 '. 

T]jat  is,  the  subtangent  on  any  diameter  of  the  parabola  is 
bisected  at  the  vertex  of  that  diameter. 

SCHOLIUM. — Projectiles,  if  not  disturbed 
by  the  resistance  of  the  atmosphere,  would 
describe  parabolas. 

Let  Pbe  the  point  from  which  a  projec 
tile  is  thrown  in  any  direction  PH.  Undis 
turbed  by  the  atmosphere  and  by  gravity,  it 
would  continue  to  move  in  that  line,  describ 
ing  equal  spaces  in  equal  times.  But  grav 
ity  causes  bodies  to  fall  through  spaces  pro 
portional  to  the  squares  of  the  times. 

From  P  draw  PL  in  the  direction  of  a  plumb  line,  the  direction 
in  which  bodies  fall  when  acted  upon  by  gravity  alone,  and  draw 
from  A,  T,  H,  etc.,  points  taken  at  pleasure  on  PH,  lines  parallel 
to  PL.  Make  AB  equal  to  the  distance  through  which  a  body 
starting  from  rest,  would  fall  while  the  undisturbed  projectile  would 
move  through  the  space  PA,  and  lay  off  TV  to  correspond  to  the 
proportion 

PA*  :  PT*::AB:  TV  (1) 

Also  lay  off  HK  to  correspond  to  the  proportion 

PA*  iPffr.AB:  EK  (2) 

In  the  same  way  we  may  construct  other  distances  on  lines  drawn 
from  points  of  PH  parallel  to  PL. 

Now  through  the  points  B}  V,  K,  etc.,  draw  parallels  to  PH, 
intersecting  PL  in  (7,  D,  L,  etc.,  and  through  the  points  B,  V, 


180  ANALYTICAL    GEOMETRY. 

Kj  etc.,  trace  a    curve.      This  curve  will  represent  the  path  de 
scribed  by  a  projectile  in  vacuo,  and  will  be  a  parabola. 

Because  AB  is  parallel  to  PC,  and  PA  parallel  to  JBC,  the  figure 
PAJSOis  a  parallelogram,  and  so  are  each  of  the  other  figures, 
PTVD,  PHKL,  etc. 

Let  PA=y,  PT=yr,  PH=y"  etc. 
and  PC=x,  PD=xf,  PL=x"  etc. 
Then  proportions  (1)  and  (2)  become  respectively 

y*  :y'*::x  :  xf 

yz  :y"*\:x  :  x" 

But  by  corollary  2  of  this  proposition,  the  curve  that  possesses 
the  property  expressed  by  these  proportions  is  the  parabola,  and  we 
therefore  conclude  that  the  path  described  by  a  projectile  in  vacuo 
is  that  curve. 


PROPOSITION   VII. 

The  parameter  of  any  diameter  of  the  parabola  is  four  times 
the  distance  from  the  vertex  of  that  diameter  to  the  focus. 
We  are  to  prove  that  2p'=4PF. 
Let  the  angle  YPR=m  as  before. 
Then  by  (Prop.  3,) 

sin.  m=rpm  a) 

cos.  w     b 

The  co-ordinates  of  the  point  P  being 
c,  by  as  in  the  last  proposition,  we  have 

b2=2pc.  (2) 

From  eq.  (1)  62sin.2ra=£>2cos.2m. 

=p2(l  —  sin  .2m)  = 

Or  sin.2m=_^  —  =  —  ^  - 

bz+p2     2pc+p2 


But  in    the    last   proposition    _    _  =2»'.    Whence 

sin.2m 

sin.  *m=-?~. 
p> 


THE   PARABOLA.  181 

Therefore  ^/— 2c+p. 

Or  2p'=4(c+£) 

\       2i  I 

But  (  c+^)  =PF.      (Prop.  1.)    Hence  2p',  the  param- 
\       ^/ 


eter  of  the  diameter  PJR,  is  four  times  the  distance  of  the 
vertex  of  the  diameter  from  the  focus. 

SCHOLIUM.— Through  the  focus  F  draw  a  line  parallel  to  the 
tangent  PY.  Designate  PR  by  x,  and  RQ  by  y.  Then,  by 
(Prop.  6), 

But  PF=FT,  (Prop.  4,  Cor.  2.)      And  PR=TF,  because 
is  a  parallelogram.     Whence  PR=PF;  and,  since  PR=x, 


and  P.#=c+, 


Therefore  4;r=4f  c-f-£-  j  =2z/.  or  <* — P 

V       2/  2 

This  value  of  a;  put  in  the  equation  of  the  curve  gives 

That  is,  the  quantity  2p',  which  has  been  called  the  parameter 
of  the  diameter  PR,  is  equal  to  the  double  ordinate  passing  through 
the  focus. 

PROPOSITION   VIII. 

If  an  ordinate  be  drawn  to  any  diameter  of  the  parabola, 
the  area  included  between  the  curve,  the  ordinate  and  the  cor 
responding  abscissa,  is  two-thirds  of  the  parallelogram  con 
structed  upon  these  co-ordinates. 

Let  V'P'PQ  be  a  portion  of  a 
parabola  included  between  the  arc 
PP'P,  and  the  co-ordinates  WQ, 
PQ  of   the  extreme  point  P,  re 
ferred  to  the  diameter  V  Q  and  the  ^ 
tangent  through  its  vertex. 
16 


182  ANALYTICAL    GEOMETRY 

Take  a  point,  P',  on  the  curve  between  P  and  V  ;  draw 
the  chord  PPf  and  the  ordinates  PQ,  Pf  Q'.  Through  N, 
the  middle  point  of  PP',  draw  the  diameter  NS,  and  at 
P  and  P'  draw  tangents  to  the  parabola  intersecting  each 
other  at  M  and  the  diameter  V7  Q  produced  at  T  and  Tf. 
The  tangents  at  the  points  P  and  Pf  have  a  common  sub- 
tangent  on  the  diameter  VS,  because  these  points,  when 
referred  to  this  diameter  and  the  tangent  at  its  vertex, 
have  the  same  abscissa,  VJN,  (Cor.  3,  Prop.  6).  The  point 
M  is  therefore  common  to  the  two  tangents  and  the  di 
ameter  VS  produced. 

By  this  construction  we  have  formed  the  trapezoid 
PQQ'P'  within,  and  the  triangle  TMT'  without,  the  par 
abola,  and  we  will  now  compare  the  areas  of  these  figures. 
From  ^draw  NL  parallel  to  PQ,  and  from  Q  draw  QO 
perpendicular  to  P'§',  and  let  us  denote  the  angle  YV'Q 
that  the  tangent  at  V  makes  with  the  diameter  V  Q  by  m. 

By  the  corollary  just  referred  to  we  have 
V  T=  V  Q  and  V  T'=  V  Q'. 

"Whence  T  T=  Qr  Q  ;  and  because  N  is  the  middle  point 
of  PP  we  also  have 


Therefore  (Th.  34,  B.  I,  Geom.,)  the  area  of  the  trap 
ezoid  PQQ  P  is  measured  by 

NLx  QO=NLx  Q'Qsin.m=Q'QxNL$m.m. 

But  NL  sin.ra  is  equal  to  the  perpendicular  let  fall  from 
j^Vupon  Qf  Q  which  is  equal  to  that  from  M  upon  the  same 
line.  Hence  the  area  of  the  triangle  TMT'  is  measured" 


The  area  of  the  trapezoid  is,  therefore,  twice  that  of 
the  triangle. 

If  another  point  be  taken  between  P'  and  V7,  and  we 
make  with  reference  to  it  and  P'  the  construction  that 


THE    PARABOLA.  183 

has  just  been  made  with  reference  to  Pf  and  P,  we  shall 
have  another  trapezoid  within,  and  triangle  without,  the 
parabola,  and  the  area  of  the  trapezoid  will  be  twice  that 
of  the  triangle. 

Let  us  suppose  this  process  continued  until  we  have  in 
scribed  a  polygon  in  the  parabola  between  the  limits  P 
and  V ;  then,  if  the  distance  of  the  consecutive  points 
P,  P',  etc.,  be  supposed  indefinitely  small,  it  is  evident 
that  the  sum  of  the  trapezoids  will  become  the  interior 
curvilinear  area  PP'V'Q,  and  the 'sum  of  the  triangles 
the  exterior  curvilinear  area  TPV  V. 

Since  any  one  of  these  trapezoids  is  to  the  correspond 
ing  triangle  as  two  is  to  one,  the  sum  of  the  trapezoids 
will  be  to  the  sum  of  the  triangles  in  the  same  propor 
tion.  But  the  interior  and  exterior  area  together  make 
up  the  triangle  PQT. 

Therefore         interior  area=f &PQT, 

and  AP§T=j:r§xP§sin.m=F'§xP§sin.w. 

But  VQxPQaiTi.  m  measures  the  area  of  the  parallel 
ogram  constructed  upon  the  abscissa  VrQ  and  the  ordi- 
nate  PQ.  We  will  denote  VQ\>jx  and  PQ  by  y.  Then 
the  expression  for  the  area  in  question  becomes 

frj/.sin.m 

Cor.  "When  the  diameter  is  the  axis  of  the  Q, 
parabola,  then  m=90°,  and  sin.  m=l,  and  the 
expression  for  the  area  becomes  fry.  That 
is,  every  segment  of  a  parabola  at  right  angles 
with  the  axis  is  two-thirds  of  its  circumscribing  rec 
tangle. 


PROPOSITION    IX. 


To  find  the  general  polar  equation  of  the  parabola. 
Let  Pbe  the  polar  point  whose  co-ordinates  referred  to 
the  principal  vertex,  T7",  are  c  and  b.    Put  VD=x,  and  D M 


184  ANALYTICAL   GEOMETRY. 

=y ;  then  by  the  equation  of  the  curve  we 
have 

y2=2px.  (1) 

Put  PM=R,  the  angle  MPJ=m,  then  y/ 
we  shall  have  \  F 

VD=x=c+ R  cos.  m. 
DM=y=b+R  sin.  m. 
These  values  of  x  and  y  substituted  in  eq.  (1)  will  give 

(b+R  sin.  m)2=2p(c+R  cos.  m).        (2) 
Expanding  and  reducing  this  equation,  (R  being  the 
variable  quantity),  we  find 

R2  sm.2m+2R(b  sin.  m— p  cos.  m)=2pc — b* 
for  the  general  polar  equation  of  the  parabola  required. 
Cor.  1.  When  P  is  on  the  curve,  b2=2pc,  and  the  gen 
eral  equation  becomes 

R2ain.*m+2R(b  sin.m — p  cos.m)=-=0. 

Here  one  value  of  R  is  0,  as  it  should  be,  and  the  other 
value  is 

•D_2(p  cos.  m — b  sin.  m) 

jfl— — i . : '- 

When  m = 270°,  cos.  m = 0  and  sin.  m= — 1.   Then  this  last 
equation  becomes 


a  result  obviously  true. 

Cor.  2.  When  the  pole  is  at  the  focus  F,  then  6=0,  and 

c=P,  and  these  values  reduce  the  general  equation  to 


But  sin.2m=l — cos.2m. 

Whence  R2 — R2cos.2m — 2Rpcos.m—p2. 

Or  R2=p2-}-2Rpeos.m+R2cos.'2m. 
Or  R=p+R  cos.m. 

Whence  R= ?. 

1 — cos.  m 

and  this  is  the  polar  equation  when  the  focus  is  the  pole. 


THE     PAEABOLA. 


185 


When  m=0,  cos.m= 
1Z=   P 


'1,  and  then  the  equation  becomes 


or 


••?.=  infinite, 
0 


1—1' 

showing  that  there  is  no  termination  of  the  curve  at  the 
right  of  the  focus  on  the  axis. 

When  m=90°,  cos.ra=0,  then  R=p,  as  it  ought  to  be, 
because  p  is  the  ordinate  passing  through  the  focus. 

When  m=180°,  cos.m= — 1,  then  R=±p;  that  is,  the 
distance  from  the  focus  to  the  vertex  is  \p. 

As  in  can  be  taken  both  above  and  below  the  axis  and 
the  cos.  m  is  the  same  to  the  same  arc  above  and  below, 
it  follows  that  the  curve  must  be  symmetrical  in  respect 
to  the  axis. 

SCHOLIUM  1. — If  we  takep  for  the  unit  of  measure,  that  is,  as 
sume  p=]-,  then  the  general  polar  equation  will  become 
J?2sin.2m-|-2JR(6sin.m— cos.m)=2c — 62. 

Now  if  we  suppose  m=90°,  then  sin.m=l,  cos.m=0,  and  R 
would  be  represented  by  the  line  PM',  and  the  equation  would  be 
come 

and  this  equation  is  in  the  common  form  of  a  quadratic. 

Hence,  a  parabola  in  which  jp=l  will  solve  any  quadratic  equa 
tion  by  making  c=  VB,  JBP=b,  then  PM'  will  give  one  value  of 
the  unknown  quantity. 

To  apply  this  to  the  solution  of  equations,  we  construct  a  parabo 
la  as  here  represented. 

Now,  suppose  we  require  the  value  of  *S 
y,  by  construction,  in  the  following  equa-  +2 
tion,  . 


Here  25=2,  and  2c— Z>2=8. 

Whence  6=1,  and  c=4.5. 

Lay  off  c  on  the  axis,  and  from  the  ex 
tremity  lay  off  b  at  right  angles,  above  the 
axis  if  b  is  plus,  and  below  if  minus. 

This  being  done,  we  find  P  is  the  polar  point  corresponding  to 
16* 


0 

-1 

-2 
-3 


1234 


M 


186  ANALYTICAL    GEOMETRY. 

this  example,  and  PMT=2  is  the  plus  value  of  y,  and  PM—  —  4  is 
the  minus  value. 

Had  the  equation  been 

j,*-2y=8,_ 

then  P'  would  have  been  the  polar  point,  and  P'M  =4  the  plus 
value,  and  P'M=  —  2  the  minus  value. 

For  another  example  let  us  construct  the  roots  of  the  following 
equation  : 

y*—  $y=—  1. 

Here  b=  —  3,  and  2c—  62=  —  7.     Whence  c=l. 

From  1  on  the  axis  take  3  downward,  to  find  the  polar  point  P". 
Now  the  roots  are  P"m  and  P"m',  both  plus.-  P"m=1.58,  and 
P"m'=4.4l4. 

Equations  having  two  minus  roots  will  have  their  polar  points 
above  the  curve. 

When  c  comes  out  negative,  the  ordinates  caiinot  meet  the  curve, 
showing  that  the  roots  would  not  be  real  but  imaginary. 

The  roots  of  equations  having  large  numerals  cannot  be  con 
structed  unless  the  numerals  are  first  reduced. 

To  reduce  the  numerals  in  any  equation,  as 


we  proceed  as  follows  : 
Puty—  nz,  then 

=  146 


n          n2 

Now  we  can  assign  any  value  to  n  that  we  please.     Suppose 
71=10,  then  the  equation  becomes 


The  roots  of  this  equation  can  be  constructed,  and  the  values  of 
y  are  ten  times  those  of  z. 

SCHOLIUM  2.  —  The  method  of  solving  quadratic  equations  em 
ployed  in  Scholium  1  may  be  easily  applied  to  the  construction  of 
the  square  roots  of  numbers. 

Thus,  if  the  square  root  of  20  were  required,  and  we  represent 
it  by  y,  we  shall  have 


THE    PARABOLA.  187 

an  incomplete  quadratic  equation;  but  it  may  be  put  under  the 
form  of  a  complete  quadratic  by  introducing  in  the  first  number  the 
term  ±  0  xy,  and  we  shall  then  have 

^±0x^=20. 

Here  25=0,  and2c—  £2=20;  whence  c=10;  which  shows  that 
the  ordinate  corresponding  to  the  abscissa  10  on  the  axis  of  the  pa 
rabola  will  represent  the  square  root  of  20.  In  the  same  way  the 
square  roots  of  other  numbers  may  be  determined 

EXAMPLES. 

1.  What  is  the  square  root  of  50  ? 

Let  each  unit  of  the  scale  represent  10,  then  50  will  be  repre 
sented  by  5.  The  half  of  5  is  2£.  An  ordinate  drawn  from  2£  on 
the  axis  of  X  will  be  about  2.24,  and  the  square  root  of  10  will  be 
represented  by  an  ordinate  drawn  from  5,  which  will  be  about  3,  16. 
Hence,  the  square  root  of  50  cannot  differ  much  from  (2.24)  (3.16) 
=  7,0786. 

ANOTHER  SOLUTION. 


50=25x2,  v/50=W2^      From  1  on  the  axis  of  X  draw  an 
ordinate  ;  it  will  measure  1.4-)-. 

Hence,  ^50=5(1.4+)=7,+. 

"What  is  the  square  root  of  175? 

175=25  x  7,  x/175  =  5v/  7. 
An  ordinate  drawn  from  3.5  the  half  of  7  will  measure  2.65. 

Therefore  ^175=5(2.65)=13.25  nearly. 

3.  Given  x2  —  T\z=8  to  find  x.     Ans.  sc=2.9.-j- 

4.  Given  %x2-\-$x=T\to&nd  x.  *  Ans.  #=0.60  -f. 

5.  Given  \y2  —  £y=2  to  find  y.     Ans.  y=3.17,  or—  2.5-f. 


188  ANALYTICAL   GEOMETRY. 

CHAPTER  V. 
THE  HYPERBOLA. 

To  describe  an  hyperbola. 

The  definition  of  this  curve  suggests  the  following 
method  of  describing  it  mechanically : 

Take  a  ruler  F'H,  and  fasten  one 
end  at  the  point  F'9  on  which  the  ru 
ler  may  turn  as  a  hinge.  At  the 
other  end  of  the  ruler  attach  a  thread, 
and  let  its  length  be  less  than  that  of 
the  ruler  by  the  given  line  A' A. 
Fasten  the  other  end  of  the  thread 
at  F. 

With  a  pencil,  P,  press  the  thread  against  the  ruler  and 
keep  it  at  equal  tension  between  the  points  H  and  F. 
Let  the  ruler  turn  on  the  point  F1 ',  keeping  the  pencil 
close  to  the  ruler  and  letting  the  thread  slide  round  the 
pencil;  the  pencil  will  thus  describe  a  curve  on  the 
paper. 

If  the  ruler  be  changed  and  made  to  revolve  about  the 
other  focus  as  a  fixed  point,  the  opposite  branch  of  the 
curve  can  be  described. 

In  all  positions  of  P,  except  when  at  A  or  A' ,  PFf  and 
PF  will  be  two  sides  of  a  triangle,  and  the  difference  of 
these  two  sides  is  constantly  equal  to  the  difference  be 
tween  the  ruler  and  the  thread ;  but  that  difference  was 
made  equal  to  the  given  line  A1  A ;  hence,  by  definition, 
the  curve  thus  described  must  be  an  hyperbola. 

PROPOSITION   I. 

To  find  the  equation  of  the  hyperbola  referred  to  its  center 

and  axes. 


THE   HTPEEBOLA.  189 


Let  C  be  the  center,  F  and  Fr  the 
foci,  and  AAr  the  transverse  axis  of 
an  hyperbola.  Draw  CO  at  right 


angles  to  AAf,  and  take  these  lines    _<s 

for  the   co-ordinate  axes.     From  P, 

any  point  of  the  curve,  draw  PF,  PFf  to  the  foci,  and 

PH  perpendicular  to  AAf. 

Make  CF=c,  CA=A,  CH=x,  and  PH=y;  then  the 
equation  which  expresses  the  relation  between  the  vari 
ables  x  and  y,  and  the  Constances  c  and  A,  will  be  the 
equation  of  a  hyperbola. 

By  the  definition  of  the  curve  we  have 

rf—  r=2A.  (1) 

The  right-angled  APffiPgives 

r*=(x—cy+y*.  (2) 

The  right-angled  &PHF'  gives 

r/a=(a;+c)a+y3.  (8) 

Subtracting  eq.  (2)  from  eq.  (3)  we  get 


Dividing  eq.  (4)  by  eq.  (1)  we  have 


. 

J. 

Combining  eqs.  (1)  and  (5)  we  find 

r'=J.+-,   and     r=—  ^L+^. 

.A  -a. 

This  value  of  r  substituted  in  eq.  (2)  gives 

A2—2cx+—=x* 

A2 
Reducing,  we  find 


for  the  equation  sought. 

SCHOLIUM.  —  As  c  is  greater  than  A,  it  follows  that  (A2  —  c2) 
must  be  negative  ;  therefore  we  may  assume  this  value  equal  to 
—  J52.  Then  the  equation  becomes 


190 


ANALYTICAL    GEOMETRY. 


This  fdrm  is  preferred  to  the  former  one  on  account  of  its  simi 
larity  to  the  equation  of  the  ellipse,  the  difference  being  only  in  the 
negative  value  of  £2. 

Because  A2—c*=—B* 


Now  to  show  the  geometrical  mag 
nitude  of  Bj  take  C  as  a  center,  and 
CF  as  a  radius,  and  describe  the  circle 
FI1F'.  From  A  draw  AH  at  right 
angles  to  CF.  Now  CH=c,  OA=A, 
and  if  we  put  AH=B,  we  shall  have  A2+B2=c2,  as  above. 
Whence  AH  must  equal  B. 


PROPOSITION   II. 


To  determine  the  figure  of  the  hyperbola  from  its  equation. 
Kesuming  the  equation 


and  solving  it  in  respect  to  y,  we  find 


If  we  make  £=0,  or  assign  to  it  any  value  less  than  A, 
the  corresponding  value  of  y  will  be  imaginary,  showing 
that  the  curve  does  not  exist  above  or  below  the  line  A' A. 


If  we  make    x=A,  then   y 
showing  two  points  in  the  curve,  both 
at  A. 

If  we  give  to  x  any  value  greater      -Pt 
than  Ay  we  shall  have  two  values  of  y, 
numerically  equal,   showing  that  the 
curve  is  symmetrically  divided  by  the  axis  Ar A  produced. 

If  we  now  assign  the  same  value  to  x  taken  negatively, 
that  is,  make  x  ( — x\  we  shall  have  two  other  values  of 
y,  the  same  as  before,  corresponding  to  the  left  branch 
of  the  curve.  Therefore,  the  two  branches  of  the  curve  are 


THE    HYPEBBOLA.  191 

equal  in  magnitude,  and  are  in  all  respects  symmetrical  but  op 
posite  in  position. 

Hence  every  diameter,  as  DD',  is  bisected  in  the  center,  for 
any  other  hypothesis  would  be  absurd. 

SCHOLIUM  1.  —  If  through  the  center,  C, 
we  draw  CD,  CD',  at  right  angles  to  A'  A, 
and  each  equal  to  J3,  we  can  have  two  opposite 
branches  of  an  hyperbola  passing  through  D 
and  D'  above  and  below  C.  as  the  two  others 
which  pass  through  the  points  A'  and  A,  at 
the  right  and  left  of  C. 

The  hyperbola  which  passes  through  D  and  D'  is  said  to  be  con 
jugate  to  that  which  passes  through  A  and  A  ',  or  the  two  are  con 
jugate  to  each  other. 

DD'  is  the  conjugate  diameter  to  A'  A,  and  DD'  may  be  less  than, 
equal  to,  or  greater  than  A'  A,  according  to  the  relative  values  of  c 
and  A  in  Prop.  1. 

When  B  is  numerically  equal  to  A,  the  equation  of  the  curve 
becomes 

y*-x*=—A*, 
and  DD'=  A  A'.    In  this  case  the  hyperbola  is  said  to  be  equilateral. 

SCHOLIUM  2.  —  To  find  the  value  of  the  double  ordinate  which 
passes  through  the  focus,  we  must  take  the  equation  of  the  curve 


and  make  x=c,  then 


But  we  have  shown  that  A*+B*—c*,  or  .B2=c2—  A*. 
Whence  A*y*=E4. 

Or  Ay=B\  or  2y=2^!. 

A 

That  is,  2  A  :  2B  :  :  2B  :  2y, 

showing  that  the  parameter  of  the  hyperbola  is  equal  to  the  double 
ordinate  t  to  the  major  axis,  that  passes  through  the  focus. 

SCHOLIUM  3.  —  To  find  the  equation  for  the  conjugate  hyper 
bola  which  passes  through  the  points  D,  Df,  we  take  the  general 
equation 


192  ANALYTICAL    GEOMETRY. 

and  change  A  into  E  and  x  into  y,  the  equation  then  becomes 

JB*x2—  A*y*=—  A*£*, 
which  is  the  equation  for  conjugate  hyperbola. 


PROPOSITION    III. 


To  find  the  equation  of  the  hyperbola  when  the  origin  is  at 
the  vertex  of  the  transverse  axis. 
When  the  origin  is  at  the  center,  the  equation  is 


And  now,  if  we  move  the  origin  to  the  vertex  at  the 
right,  we  must  put 


Substituting  this  value  of  x  in  the  equation  of  the   hy 
perbola  referred  to  its  center  and  axes,  we  have 
Ay—B*x  '2—2JB2Ax  '=  0. 

We  may  now  omit  the  accents,  and  put  the  equation 
under  the  following  form  : 


which  is  the  equation  of  the  hyperbola  when  the  origin 
is  the  vertex  and  the  co-ordinates  rectangular. 

PROPOSITION   IY. 

To  find  the  equation  of  a  tangent  line  to  the  hyperbola,  the 
origin  being  the  center. 

In  the  first  place,  conceive  a  line 
cutting  the  curve  in  two  points,  P 
and  Q.    Let  x  and  y  be  co-ordinates 
of  any  point  on  the  line,  as  $,  xr 
and  y'  co-ordinates  of  the  point  P 
on  the  curve,  and  x"  and  y"  the  co-       s/ 
ordinates  of   the  point   Q  on  the — £. 
curve. 


THE    HYPERBOLA.  193 

The  student  can  now  work  through  the  proposition  in 
precisely  the  same  manner  as  Prop.  6,  of  the  ellipse  was 
worked,  using  the  'equation  for  the  hyperbola  in  place  of 
that  of  the  ellipse,  and  in  conclusion  he  will  find 


for  the  equation  sought. 

Cor.  To  find  the  point  in  which  a 
tangent  line  cuts  the  axis  of  JL,  we 
must  make  y=0,  in  the  equation  for 
the  tangent ;  then 


x 

If  we  subtract  this  from  CD  (xf)  we  shall  have  the  sub 
tangent  =—A*    x'2—A* 

~ 


PRO  POSITION   Y. 

To  find  the  equation  of  a  normal  to  the  hyperbola. 

Let  a  be  the  tangent  of  the  angle  that  the  line  TP  makes 

with  the  transverse  axis,  (see  last  figure),  and  a'  the  same 

with  reference  to  the  line  PN.     Then  if  PN  is  a  normal, 

it  must  be  at  right  angles  to  PT,  and  hence  we  must  have 

oa'+l=0.  (1) 

Let  x'  and/  be  the  cor-ordinates  of  the  point  P  on  the 
curve,  and  x,  ?/,  the  co-ordinates  of  any  point  on  the  line 
PN,  then  we  must  have 

y-.yr=af(x—  x'}.  (2) 

In  working  the  last  proposition,  for  the  tangent  line 
PTwe  should  have  found 


This  value  of  a  put  in  eq.  (1)  will  show  us  that 
a!~ 

17 


194  ANALYTICAL    GEOMETRY. 

And  this  value  of  a'  put  in  eq.  (2)  will  give  ns 


for  the  equation  of  the  normal  required. 

Cor.  To  find  the  point  in  which  the  normal  cuts  the 
axis  of  X,  we  must  make  y=0. 

This  reduces  the  equation  to 


Whence 


=  CN. 


If  we  subtract  CD,  (x')7  from  CN9  we  shall  have  DN, 
the  sub-normal. 


That  is, 


,  the  sub-normal. 


PROPOSITION  VI. 


A  tangent  to  the  hyperbola  bisects  the  angle  contained  by  lines 
drawn  from  the  point  of  contact  to  the  foci. 

If  we  can  prove  that 

F'P:  PF:  :F'T:TF,     (1) 
it  will  then  follow  (Th.  24,  B.  IE, 
Geom.,)  that  the  angle  F'PT=t~hQ 
angle  TPF. 

In  Prop.  1,  of  the  hyperbola,  we 
find  that 


=r'=^L+.-,  and 
A 

and  by  corollary  to  Prop.  4 
F>T=FfC+CT=c+ 

We  will  now  assume  the  proportion 


—,  and  TF= 

x 


x 


THE   HYPERBOLA.  195 

Multiply  the  terms  of  the  first  couplet  by  A,  and  those 
of  the  last  couplet  by  x,  then  we  shall  have 

(A*  +  cx) :  (— A*+cx)  :  :  (cx+A2)  :  xz. 
Observing  that  the  first  and  third  terms  of  this  propor 
tion  are  equal,  therefore 

xz—cx — A2. 

Or  z=c— ^-=TF.- 

x 

Now  the  first  three  terms  of  proportion  (2)  were  taken 
equal  to  the  first  three  terms  of  proportion  (1),  and  we 
have  proved  that  the  fourth  term  of  proportion  (2)  must 
be  equal  to  the  fourth  term  of  proportion  (1),  therefore 
proportion  (1)  is  true,  and  consequently 
F'PT=TPF. 

Cor.  1.  As  TT  is  a  tangent,  and  PN  its  normal,  it 
follows  that  the  angle  TPN=  the  angle  FPN,  for  each 
is  a  right  angle.  From  these  equals  take  away  the  equals 
TPF,  T'PQ,  and  the  remainder  FPNwuti  equal  the  re 
mainder  QPN.  That  is,  the  normal  line  at  any  point  of  the 
hyperbola  bisects  the  exterior  angle  formed  by  two  lines  drawn 
from  the  foci  to  that  point. 

A  2 

Cor.  2.  The  value  of  CT  we  have  found  to  be   — ,  and 

x 

the  value  of  CD  is  x,  and  it  is  obvious  that 
A*  •  A  •  •  A  •  x 

.   ~cL  .    .  yi    .X, 

X 

is  a  true  proportion.  Therefore  (A)  is  a  mean  proportional 
between  CT  and  CD. 

A  tangent  line  can  never  meet  the  axis  in  the  center, 
because  the  above  proportion  must  always  exist,  and  to 
make  the  first  term  zero  in  value,  we  must  suppose  x  to  be 
infinite.  Therefore  a  tangent  line  passing  through  the  center 
cannot  meet  the  hyperbola  short  of  an  infinite  distance  there 
from. 

Such  a  line  is  called  an  asymptote. 


196 


ANALYTICAL   GEOMETRY. 


OF  THE  CONJUGATE  DIAMETERS  OF  THE  HYPERBOLA. 

DEFINITION.  —  Two  diameters  of  an  hyperbola  are  said  to 
be  conjugate  when  each  is  parallel  to  a  tangent  line  drawn 
through  the  vertex  of  the  other. 

According  to  this  definition,  GGr'  and  jET.fi"'  in  the  ad 
joining  figure  are  conjugate  diameters. 

EXPLANATION.  1.  —  The  tangent  line 
which  passes  through  the  point  H  is  par 
allel  to  CG.  Hence  CG  makes  the  same 
angle  with  the  axis  as  that  tangent  line 
does. 

If  we  designate  the  co-ordinates  of  the 
point  If,  in  reference  to  the  center  and  axes 
by  x'  and  y',  and  by  a  the  tangent  of  the 
angle  made  by  the  inclination  of  CG  with  the  axis,  then  in  the  in 
vestigation  (Prop.  6,)  we  find 


Now  if  we  designate  the  tangent  of  the  angle  which   CH  makes 
with  the  axis  by  a',  the  equation  of  OH  must  be  of  the  form 


because  the  line  passes  through  the  center. 
Whence  a'  •=.  __ . 


(2) 


Multiplying  eqs.  (1)  and  (2)  together  member  by  member,  and 
we  find 


to  which  equation  all  conjugate  diameters  must  correspond. 

EXPLANATION  2. — If  we  designate  the  angle  GOB  by  n,  and 
HOB  by  m,  we  shall  have 


And 


sin.  m       ,       sin.  n 
=a  ,     =a. 

cos.  m              cos.  n 
tan.  m  tan.  n= , 


THE   HYPERBOLA.  197 

PROPOSITION    VII. 

To  find  the  equation  of  the  hyperbola  referred  to  its  center 
and  conjugate  diameters. 

The  equation  of  the  curve  referred  to  the  center  and 
axes  is 


,  to  change  rectangular  co-ordinates  into  oblique, 
the  origin  being  the  same,  we  must  put 

x~*f  cpB.m+y'cpB.n  } 
And  y=xf  sin.  m+y'  sin.  n  J 

These  values  of  x  and  y,  substituted  in  the  above  gen 
eral  equation,  will  produce 


-f  2(sin.  m  sin.  nA*—  cos.  m  cos.  n 
=—  A*£*.  (1) 

Because  the  diameters  are  conjugate,  we  must  have 
sin.  m     sin.  n    B2 
cos.  m    cos.  n    A2 

Whence    (sin.  m  sin.  n  A2  —  ccs.  m  cos.  nB2)=Q     (k) 
This  last  equation  reduces  eq.  (1)  to 


which  is  the  equation  of  the  hyperbola  referred  to  the 
center  and  conjugate  diameters. 

If  we  make  2/r=0,  we  shall  have 


(3) 


If  we  make  x'=0,  we  shall  have 

J2J?2 

-,=CG        (4) 


If  we  put  A'2  to  represent  CH*,  and  regard  it  as  posi 
tive,  the  denominator  in  eq.  (3)  must  be  negative,  the  nu- 


198  ANALYTICAL    GEOMETRY. 

merator  being  negative.      That  is,  ^2sin.2m  must  be  less 
than  J92cos.2w. 

That  is,  .42sin.2ra<£2cos.2w. 

B 

tan.  ra<-7. 

7>a 

B  ut  tan.  m  tan.  n  —  -p. 

T) 

Whence  tan.  n>-f,  °r?  ^2sin.2ft>.52cos.2%. 

Therefore  the  denominator  in  eq.  (4)  is  positive,  but 
the  numerator  being  negative,  therefore  CG*  must  be 
negative.  Put  it  equal  to  —  B". 

Now  the  equations  (3)  and  (4)  become 

A'*  = j-^-^T— ? f ,  —  B*—  -  a  .    a^2-a 

Or  (^2sin.2m 


Comparing  these  equations  with  eq.  (2)  we  perceive 
that  eq.  (2)  may  be  written  thus  : 


"Whence  A'*yn 

Omitting  the  accents  of  xf  and  y*  ',  since  they  are  gene 
ral  variables,  we  have 

A'y—Bf2x2=—A'2£'2, 

for  the  equation  of  the  hyperbola  referred  to  its  center 
and  conjugate  diameters. 

SCHOLIUM  1.  —  As  this  equation  is  precisely  similar  to  that  re 
ferred  to  the  center  and  axes,  it  follows  that  all  results  hitherto  de 
termined  in  respect  to  the  latter  will  apply  to  conjugate  diameters 
by  changing  A  to  A'  and  B  to  B', 

For  instance,  the  equation  for  a  tangent  line  in  respect  to  the 
center  and  axes  has  been  found  to  be 


THE   HYPERBOLA. 
Therefore,  in  respect  to  conjugate  diameters  it  must  be 


199 


and  so  on  for  normals,  sub-normals,  tangents  and  sub-tangents. 
SCHOLIUM  2,  —  If  we  take  the  equation 


and  resolve  it  in  relation  to  y,  we  shall  find 
that  for  every  value  of  x  greater  than  Ar  we 
shall  find  two  values  of  y  numerically  equal, 
which  shows  that  ON  bisects  MM  and  every 
line  drawn  parallel  to  MM,  or  parallel  to  a 
tangent  drawn  through  L,  the  vertex  of  the 
diameter  LL'. 

Let  the  student  observe  that  these  several  geometrical  truths  were 
discovered  by  changing  rectangular  to  oblique  co-ordinates.  We 
will  now  take  the  reverse  operation,  in  the  hope  of  discovering  other 
geometrical  truths. 

Hence  the  following  : 


PROPOSITION    VIII. 

To  change  the  equation  of  the  hyperbola  in  reference  to  any 
system  of  conjugate  diameters,  to  its  equation  in  reference  to  the 
axes. 

The  equation  of  the  hyperbola  referred  to  conjugate 
diameters  is 

A'2y'2—_B'2xf2=—A'  IS'2. 

To  change  oblique  to  rectangular  co-ordinates,  the  for 
mulas  are  (Chap.  1,  Prop.  10,) 

,_xain.n — ?/cos.n  ,_?/cos.w — xsin.  m 

sin.  (n — m)     '  sin.  (n — m) 

Substituting  these  values  of  x'  and  yf  in  the  equation, 
we  shall  have 

A'\y  cos.  m — x  sin.  m)2 B'\x  sin.  n — y  cos.  rif_      j,2  M 

sin.F(w— m)  sin.2(n— m) 

By  expanding  and  reducing,  we  shall  have 


200 


ANALYTICAL    GEOMETRY. 


(J./2cos.2w— 

2(  — 

=—  A2Bf2  sin.2(ft—  m). 

which,  to  be  the  equation  of  the  hyperbola  when  referred 
to  the  center  and  axes,  must  take  the  well  known  form, 

A2f—B2x2=—A2B2. 

Or  *in  other  words,  these  two  equations  must  be,  in 
fact,  identical,  and  we  shall  therefore  have 

A'2  cos.2m—B'2  co82n=A2.  (1) 

A'2  8in.2m—B'2  sin.2/*=—  B2.  (2) 

—  An  sin.  ra  cos.  m-\-Bf2  sin  ,  n  cos.  n=  0.  (3) 

—  A'2  Bfz  $in.\n—m)=—A2B2.  (4) 

By  adding  eqs.  (1)  and  (2),  observing  that  (eos.2w-J- 
8in.2m)=l,  we  shall  have 


Or  4J/2—  4Bf2=4Ay 

which  equation  shows  this  general  geometrical  truth  : 
That  the  difference  of  the  squares  of  any  two  conjugate  di 

ameters  is  equal  to  the  difference  of  the  squares  of  the  axes. 
Hence,  there  can  be  no  equal  .conjugate  diameters  un 

less  A—By  and  then  every  diameter  will  be  equal  to  its  con 

jugate  :  that  is,  A'—B'. 

T>2 

Equation  (3)  corresponds  to  tan.mtan.n=  _  ?  the  -equa- 

_^L 

tion  of  condition  for  conjugate  di 
ameters. 

Equation  (4)  reduces  to 

A'B  &m.(n—m)=AB. 

The  first  member  is  the  measure 
of  the  parallelogram  GCHT,  and  it 
being  equal  to  A  X  B,  shows  this  ge 
ometrical  truth  : 

That  the  parallelogram  formed  by  drawing  tangent  lines 
through  the  vertices  of  any  system  of  conjugate  diameters  of 


THE    HYPERBOLA.  201 

the  hyperbola,  is  equivalent  to  the  rectangle  formed  by  drawing 
tangent  lines  through  the  vertices  of  the  axes. 

REMAKK. — The  reader  should  observe  that  this  propo 
sition  is  similar  to  (Prop.  13,)  of  the  ellipse,  and  the  gen 
eral  equation  here  found,  and  the  incidental  equations  (1), 
(2),  (3),  and  (4),  might  have  been  directly  deduced  from 
the  ellipse  by  changing  B  into  B  */  i?  and  Br  into 


OF  THE  ASYMPTOTES  OF  THE  HYPERBOLA. 

DEFINITION. — If  tangent  lines  be  drawn  through  the 
vertices  of  the  axes  of  a  system  of  conjugate  hyperbolas, 
the  diagonals  of  the  rectangle  so  formed,  produced  inde 
finitely,  are  called  asymptotes  of  the  hyperbolas. 

Let  AA',  BB',  be  the  axes  of 
conjugate  hyperbolas,  and  through 
the  vertices  A ,  A',  B,  B' ,  let  tan 
gents  to  the  curves  be  drawn  form 
ing  the  rectangle,  as  seen  in  the 
figure.  The  diagonals  of  this  rect 
angle  produced,  that  is,  DDf  and 
EE' ,  are  the  asymptotes  to  the  curves  corresponding  to 
the  definition. 

If  we  represent  the  angle  -DOJTby  m,  E'  CX  will  be  m 
also,  for  these  two  angles  are  equal  because  CB=  CB'. 

It  is  obvious  that 

T> 

tan.  m— 

A 

sin.2  m    B2 

whence  2 = -^ 

cos2,  m      A2 

But  cos.2  m=l — sin.^ra.     Therefore 

sin.2m    _-B2 
•    1— sin.2m     A2 


202  ANALYTICAL    GEOMETRY. 

Consequently  sin.2  m=     2     P2>  and  cos.2  w=-_ __, 

.4.   ~T -D  -A  ~T-O 

which  equations  furnish  the  value  of  the  angle  which  the 
asymptotes  form  with  the  transverse  axis. 

PROPOSITION    IX. 

To  find  the  equation  of  the  hyperbola,  referred  to  its  center 
and  asymptotes. 

Let  CM=x,  and  PM—y.     Then  the  equation  of  the 
curve  referred  to  its  center  and  axes  is 

— A*&.  (1) 


From  P  draw  PH  parallel  to  CE,  and 
PQ  parallel  to   CM.    Let  CH=x',  and 


~Now  the  object  of  this  proposition  is 
to  find  the  values  of  x  and  y  in  terms  of 
x'  and?/',  to  substitute  them  in  eq.  (1). 
The  resulting  equation  reduced  to  its 
most  simple  form  will  be  the  equation 
sought. 

The  angle  HCMis  designated  by  m,  and  because  IIP 
is  parallel  to  CE,  and  PQ  parallel  to  CM,  the  angle  HPQ 
is  also  equal  to  m. 

Now  in  the  right  angled  triangle  CHh  we  have  Hh 
=x'  sin.  m,  and  Ch—x'  cos.  m. 

In  the  right  angled  triangle  PQH  we  have  HQ 
=yr  sin.  m,  and  PQ=y'  cos.  m. 

Whence  Hh  —  HQ=Qh=PM=y=x'  sin.  m  —  yr  sin.  m. 

Or  y=(%'  —  j/0  sin.  m.  (2) 

Ch+  QP=  CM=x=xf  cos.  m-f  y'  cos.  m. 
Or  z=(a:'-HO  cos.  m.  (3) 

These  values  of  y  and  x  found  in  eqs.  (2)  and  (3)  sub 
stituted  in  eq.  (1)  will  give 


THE   HYPERBOLA.  203 


A\x'—yJ  sin.2  ra—  W  (x'+y'}2  cos.2  w=—  J.2JB2. 

Placing  in  this  equation  the  values  of  sin.2mandcos.2w, 
previously  determined,  we  have 


Dividing  through  by  AZB2,  and  at  the  same  time  mul 
tiplying  by  (J.2-f  -B2),  we  get 


Or 

or  . 

which  is  the  equation  of  the  hyperbola  referred  to  its 
center  and  asymptotes. 

Cor.  As  x'  and  yf  are  general  variables,  we  may  omit 
the  accents,  and  as  the  second  member  is  a  constant 
quantity,  we  may  represent  it  by  M2.  Then 

xy=  M2,  or  x=^l 

y 

This  last  equation  shows  that  x  increases  as  y  decreases  ; 
that  is,  the  curve  approaches  nearer  and  nearer  the  asymptote 
as  the  distance  from  the  center  becomes  greater  and  greater. 

But  x  can  never  become  infinite  until  y  becomes  0;  that 
is,  the  asymptote  meets  the  curve  at  an  infinite  distance,  corres 
ponding  to  Cor.  2,  Prop.  6. 

PROPOSITION    X. 

All  parallelograms  constructed  upon  the  abscissas,  andordi- 
nates  of  the  hyperbola  referred  to  its  asymptotes  are  equivalent, 
each  to  each,  and  each  equivalent  to  JAB. 

Let  x  and  y  be  the  co-ordinates  corresponding  to  any 
point  in  the  curve,  as  P.  Then  by  the  equation  of  the 
curve  in  relation  to  'the  center  and  asymptotes,  we  have 

x=Mz.  (1) 


204  ANALYTICAL  GEOMETRY. 

Also  let  x',  y'9  represent  the  co-ordinates 
of  the  point  Q.     Then 

x'y'=M\  (2) 

The  angle  p  CD  between  the  asymptotes 
we  will  represent  by  2m.  Now  multiply 
both  members  of  equations  (1)  and  (2)  by 
sin.  2m. 

Then  we  shall  have 

xy  sin.  2m=  M2  sin.  2m.  (3) 

x'y1  sin.  2m=  M2  sin.  2m.  (4) 

The  first  member  of  eq.  (3)  represents  the  parallelo 
gram  (7P,  and  the  first  member  of  eq.  (4)  represents  the 
parallelogram  CQ  ;  and  as  each  of  these  parallelograms 
is  equivalent  to  the  same  constant  quantity,  they  are  equiv 
alent  to  each  other. 

Now  A  is  another  point  in  the  curve,  and  therefore  the 
parallelogram  AH  CD  is  equal  to  (M2  sin.  2m),  and  there 
fore  equal  to  CQ,  or  CP.  Hence  all  parallelograms 
bounded  by  the  asymptotes  and  terminating  in  a  point  in 
the  curve,  are  equivalent  to  one  another,  and  each  equiv 
alent  to  the  parallelogram  AHCD,  which  has  for  one  of 
its  diagonals  half  of  the  transverse  axis  of  A. 

We  have  now  to  find  the  analytical  expression  for  this 
parallelogram. 

The  angle  HCA=m,  ACD=m,  and  because  AH  is  pa 
rallel  to  <7D,  CAH=m.  Hence,  the  triangle  CAH  is 
isosceles,  and  CH=HA.  The  angle  AHq=2m.  Now 
by  trigonometry 

sin.  2m  :  A  :  :  sin.  m  :  CH. 
But  sin.  2m=2  sin.  m  cos.  m.    Whence 

2  sin.  m  cos.  m  :  A  :  :  sin.  m  :  CH. 


2cos.m 

Multiply  each  member  of  this  equation  by  CA—  A9  and 
sin.m,  then 


THE   HYPERBOLA. 


205 


.  m. 


2   cos.m     2 
The  first  member  of  this  equation  represents  the  area 

75 

of  the  parallelogram  CHAD,  and  the  tan.  w=_.      Hence, 

A. 

A2    T> 

the  parallelogram  is  equal  _-_=J^-B,  which  is  the  value 

2t  A. 

also  of  all  the  other  parallelograms,  as  CQ,  CP,  etc. 


PROPOSITION   XI. 

To  find  the  equation  of  a  tangent  line  to  the  hyperbola  re 
ferred  to  its  center  and  asymptotes. 

Let  P  and  Q  be  any  two  points  on  the 
curve,  and  denote  the  co-ordinates  of  the 
first  by  x',  #',  and  of  the  second  by  #",  y". 

The  equation  of  a  straight  line  pass 
ing  through  these  points  will  be  of  the 
form 


y—  y'=a(x—  x') 


(1) 


in  which  a=^     ^  . 
x'—  x" 

We  are  now  to  find  the  value  of  a  when  the  line  be 
comes  a  tangent  at  the  point  P. 
Because  P  and  Q  are  points  in  the  curve,  we  have 
xfy'=x"y". 

From  each  member  of  this  last  equation  subtract 
then 


Or 

Whence 


x'(y'-y")=-y»(x'-x"). 
=^     *  —     ** 


x'—x\   '    x'' 
This  value  of  a  put  in  eq.  (1)  gives 

y—  /=—  ^(x-xf). 


(2) 


ANALYTICAL    GEOMETRY. 

Now  if  we  suppose  the  line  to  revolve  on  the  point  P 
as  a  center  until  Q  coincides  with  P,  then  the  line  will 
be  a  tangent,  and  xf=x",  and  y'=y",  and  eq.  (2)  will  be 
come 

?/ 

y— y'=— ?-(x— *')> 

which  is  the  equation  sought. 

Cor.  To  find  the  point  in  which  the 
tangent  line  meets  the  axis  of  JT,  we 
must  make  y=Q  ;  then 
x=2x'. 

That  is,  Ct  is  twice  CR,  and  as  RP 
and  CT  are  parallel,  tP=PT. 

A  tangent  line  included  between  the  asymp 
totes  is  bisected  by  the  point  of  tangency. 

SCHOLIUM. — From  any  point  on  the  asymptote,  as  Z>,  draw  D  G 
parallel  to  Tt,  and  from  C  draw  CP,  and  produce  it  to  S. 

By  scholium  2  to  Prop.  7  we  learn  that  CP  produced  will  bisect 
all  lines  parallel  to  tT  and  within  the  curve;  hence  gd  is  bisected 
in  S. 

But  as  CP  bisects  tT,  it  bisects  all  lines  parallel  to  tT  within  the 
asymptotes,  and  DG  is  also  bisected  in  S ;  hence  dD=  Gg. 

In  the  same  manner  we  might  prove  dh=kv,  because  hk  is  par 
allel  to  some  tangent  which  might  be  drawn  to  the  curve,  the  same 
as  D  G  is  parallel  to  the  particular  tangent  t  T. 

Hence,  If  any  line  be  drawn  cutting  the  hyperbola,  the  parts  be 
tween  the  asymptotes  and  the  curve  are  equal. 

This  property  enables  us  to  describe  the  hyperbola  by  points,, 
when  the  asymptotes  and  one  point  in  the  curve  are  given. 

Through  the  given  point  d,  draw  any  line,  as  D  G,  and  from  G 
set  off  Gg=dD,  and  then  g  will  be  a  point  in  the  curve.  Draw 
any  other  line,  as  hk,  and  set  off  kv=dh  ;  then  v  is  another  point 
in  the  curve.  And  thus  we  might  find  other  points  between  v  and 
g,  or  on  either  side  of  v  and  g. 


THE   HYPERBOLA.  207 

PROPOSITION    XII. 

To  find  the  polar  equation  of  the  hyperbola,  the  pole  being  at 
either  focus. 

Take  any  point  P  in  the  hyperbola, 
and  let  its  distance  from  the  nearest 
focus  be  represented  by  r,  and  its  dis 
tance  from  the  other  focus  be  repre-   p>  A:  C    A!    F  H 
sented  by  r'. 

Put  CH=x,  OF=c,  and  CA=A.  Then,  by  Prop.  1, 
we  have 

™,  (1) 

A. 

™  (2) 

A    * 

"Now  the  problem  requires  us  to  replace  the  symbol  x, 
in  these  formulas,  by  its  value,  expressed  in  terms  of  r 
and  r',  and  some  function  of  the  angle  that  these  lines 
make  with  the  transverse  axis. 

First. — In  the  right-angled  triangle  PFH,  if  we  desig 
nate  the  angle  PFH  by  v,  we  shall  have 

1  :  r  :  :  cos.  v  :  FH=r  cos.  v. 
CH=  CF+FH.    That  is,  x=c+r  cos.  v. 
The  value  of  x  put  in  eq.  (1)  gives 
A.(?+cr  cos.  v 
—T 

Whence  r=    ^—^  _.  (3) 

A — c  cos.  v 

Second. — In  the  right-angled  triangle  F'PH,  if  we  des 
ignate  the  angle  PF'H^j  vf,  we  shall  have 
1  :  r'  :  :  cos.  vf  :  FfH=r'  cos.  v'. 
But    F'H=F'  0+  CH.     That  is,  r'  cos.  v'=c+x. 
Or  x=r'  cos.  vr — c* 


208  ANALYTICAL    GEOMETKY. 

and  this  value  of  x  put  in  eq.  (2)  gives 

crr  cos.  vf — (? 


Whence  r'=  . 

A  —  c  cos.  v1 

Equations  (3)  and  (4)  are  the  polar  equations  required. 
Let  us  examine  eq.  (3).     Suppose  fl=0,  then  cos.v=l, 
and 


A—c 

But  a  radius  vector  can  never  be  a  minus  quantity, 
therefore  there  is  no  portion  of  the  curve  on  the  axis  to 
the  right  of  F. 

To  find  the  length  of  r  when  it  first  strikes  the  curve, 
we  find  the  value  of  the  denominator  when  its  value  first 
becomes  positive,  which  must  be  when  A  becomes  equal 
to  c  cos.  v  ;  that  is,  when  the  denominator  is  0.  the  value 
of  r  will  be  real  and  infinite. 

If  A  —  ccos.v=0, 

A 
then  cos.  v—  _  . 

c 

This  equation  shows  that  when  r  first  meets  the  curve 
it  is  parallel  to  the  asymptote,  and  infinite. 

When  v=90°,  cos.v=0,  and  then  r  is  perpendicular  at 

the  point  F,  and  equal  to  c     _   or  __  half  the  paranie- 

A  A 

ter  of  the  curve,  as  it  ought  to  be. 

"When  #=180°,  then  cos.v=  —  1,  and  —  ccos.v=c;  then 


c+A 
a  result  obviously  true. 

As  v  increases,  the  value  of  r  will  remain  positive,  and, 
consequently,  give  points  of  the  hyperbola  until  cos.-y 

again  becomes'equal  to  _  ,  which  will  be  when  the  radius 

c 


THE   HYPERBOLA.  209 

vector  makes  with  the  transverse  axis  an  angle  equal  to 

360°  minus  that  whose  cosine  is  __.     Equation   (3)  will 

c 

therefore  determine  all  points  in  the  right  hand  branch 
of  the  hyperbola. 

Now  let  us  examine  equation  (4).     If  we  make  i/=0, 
then 


A  —  c 
as  it  ought  to  be. 

To  find  when  rf  will  have  the  greatest  possible  value, 
we  must  put 

A  —  ccos.?/=0. 

Whence  cos.t/=  __  . 

c 

This  shows  that  v'  is  then  of  such  a  value  as  to  make  r1 
parallel  to  the  asymptote  and  infinite  in  length.  If  we 
increase  the  value  of  v'  from  this  point,  the  denominator 
will  become  positive,  while  the  numerator  is  negative, 
which  shows  that  then  /  will  become  negative,  indicating 
that  it  will  not  meet  the  curve. 

The  value  of  r  will  continue  negative  until  the  radius 
vector  falls  below  the  transverse  axis,  and  makes  with  it 

an  angle  having  -f  _  for  its  cosine.     Values  of  v  between 

c  . 

this  and  360°  will  render  r  positive  and  give  points  of  the 
hyperbola.  Equation  (4)  will,  therefore,  also  determine 
all  the  points  in  the  right  hand  branch  of  the  hyperbola-. 
By  changing  the  sign  of  c,  we  change  the  pole  to  the 
focus  F'  ,  and  eqs.  (3)  and  (4),  which  then  determine  the 
left  hand  branch  of  the  hyperbola,  become 

(3') 


.  v 
and  r^    A*—  c*  „   (4/) 

A+c  cos.  v' 
10*  o 


210  ANALYTICAL    GEOMETRY. 

GENERAL  REMARKS.  —  When  the  origin  of  co-ordinates  is  at  the 
circumference  of  a  circle,  its  equation  is 


When  the  origin  of  a  parabola  is  at  its  vertex,  its  equation  is 

ya=2pa;. 

When  the  origin  of  co-ordinates  of  the  ellipse  is  at  the  vertex  of 
the  major  axis,  the  equation  of  the  curve  is 


When  the  origin  of  co-ordinates  is  at   the  vertex  of  the  hyper 
bola,  the  equation  for  that  curve  is 


But  all  of  these  are  comprised  in  the  general  equation 


In  the  circle  and  the  ellipse,  q  is  negative  ;  in  the  hyperbola  it  is 
positive,  and  in  the  parabola  it  is  0. 


CHAPTER  VI. 

ON  THE  GEOMETRICAL  REPRESENTATION  OF  EQUATIONS 
OF  THE  SECOND  DEGREE  BETWEEN  TWO  YARIABLES. 

1. — It  has  been  shown  in  Chap.  1,  that  every  equa 
tion  of  the  first  degree  between  two  variables  may  be 
represented  by  a  straight  line. 

It  has  also  been  shown  that  the  equations  of  the  circle, 
the  ellipse,  the  parabola  and  the  hyperbola  were  all  some 
of  the  different  forms  of  an  equation  of  the  second  de 
gree  between  two  variables.  It  is  now  proposed  to  prove 
that,  when  an  equation  of  the  second  degree  between  two 
variables  represents  any  geometrical  magnitude,  it  is 
some. one  of  these  curves. 

The  limits  assigned  to  this  work  compel  us  to  be  as 
brief  in  this*  investigation  as  is  consistent  with  clearness. 
"We  shall,  therefore,  restrict  ourselves  to  a  demonstration 


INTERPRETATION   OF  EQUATIONS.     211 

of  this  proposition  ;  the  determination  of  the  criteria  by 
which  it  may  be  decided  in  every  case  presented,  to  which 
of  the  conic  sections  the  curve  represented  by  the  equa 
tion  belongs,  and  the  indication  of  the  processes  by 
which  the  curve  may  be  constructed. 

2. — The  equation  of  the  second  degree  between  two 
variables,  in  its  most  general  form,  is 

Ay*+Bxy+  Cx*+Dy+Ex+F=Q, 

for,  by  giving  suitable  values  to  the  arbitrary  constants, 
A9  B,  Cj  etc.,  every  particular  case  of  such  equation  may 
be  deduced  from  it. 

The  formulas  for  the  transformation  of  co-ordinates 
being  of  the  first  degree  in  respect  to  the  variables,  the 
degree  of  an  equation  will  not  be  changed  by  changing 
the  reference  of  the  equation  from  one  system  of  co-or 
dinate  axes  to  another.  We  may  therefore  assume  that 
our  co-ordinate  axes  are  rectangular  without  impairing 
the  generality  of  our  investigation. 

The  resolution,  in  respect  to  ?/,  of  the  general  equation 
gives 


~2AX    2^L±2^\|. 


—4AC 

"Now  let  A X,  A  Y  be  the  co-ordinate  axes,  and  draw 
the  straight  line  M  Q,  whose  equation  is 

_B_X__I>_ 

2  A       2  A* 

For  any  value,  AD,  of  x,  the  or- 
dinate,  DC,  of  this  line,  is  ex 
pressed  by 


B    D      E      X 


2T        2T' 

AA  ^J^l.  x 

and  this  ordinate,  increased  and  diminished  successively 
by  what  the  radical  part,  when  real,  of  the  general  value 
of  y  becomes  for  the  same  substitution  for  x,  will  give 


212  ANALYTICAL    GEOMETRY. 

two  ordinates;  DP,  DP',  corresponding  to  the  abscissa 
AD. 

Since  P  and  P  are  two  points  whose  co-ordinates, 
when  substituted  for  x  and  ?/,  will  satisfy  the  equation, 
Ay*+Bxy+Gx?+,  etc.,  =0,  they  are  points  in  the  line 
that  this  equation  represents.  By  thus  constructing  the 
values  of  y  answering  to  assumed  values  of  x,  we  may 
determine  any  number  of  points  in  the  curve. 

In  getting  the  points  P  and  P',  we  laid  off,  on  a  par 
allel  to  the  axis  of  y,  equal  distances  above  and  below 
the  point  C;  PP'  is,  therefore,  a  chord  of  the  curve  par 
allel  to  that  axis,  and  is  bisected  at  the  point  C. 

The  solution  of  the  general  equation  in  respect  to  x, 
gives 


- 

2(7"     2(7 


—  4GP  —  4  OF 


The  equation 


_ 
20       2C" 

is  that  of  a  straight  line,  making,  with  the  axis  of  #,  an 

T) 

angle  whose  tangent  is  —  _  ,  and  intersecting  the  axis 

2t  G 

-rji 

of  JT  at  a  distance  from  the  origin  equal  to  —  -  . 

A  G 

As  above,  it  may  be  shown  that  any  value  of  y  that 
makes  the  radical  part  of  the  general  value  of  x  real,  re 
sponds  to  two  points  of  the  curve,  and  that  the  straight 
line  whose  equation  is 

x^-Zy-Z 
IC2'    20' 

bisects  the  chord,  parallel  to  the  axis  of  Jf,  that  joins 
these  points. 

By  placing  the  quantity  under  the  radical  sign  in  the 
value  of  y  equal  to  0,  we  have  an  equation  of  the  second 
degree  in  respect  to  #,  which  will  give  two  values  for  x. 


INTEKPRETATION   OF   EQUATIONS.     213 

If  these  values  are  real  the  corresponding  points  of  the 
curve  are  on  the  line  M  Q  ;  that  is,  they  are  the  intersec 
tions  of  this  line  with  the  curve,  since,  for  each  of  these 
values,  there  will  be  but  one  value  of  y,  which,  in  con 
nection  with  that  of  x,  will  satisfy  the  general  equation, 
and  these  values  also  satisfy  the  equation, 


2A       2JL 

In  like  manner,  placing  the  quantity  under  the  radical 
sign  in  the  value  of  x  equal  to  0,  we  shall  find  two  values 
of  y,  to  each  of  which  there  will  respond  a  single  value 
of  x,  and  the  points  of  the  curve  answering  to  these  val 
ues  of  y  will  be  the  intersections  of  the  curve  with  the 
line  whose  equation  is 

*=-_V-_^ 

2(7      2(7 

A  diameter  of  a  curve  is  defined  to  be  any  straight  line 
that  bisects  a  system  of  parallel  chords  of  the  curve. 
From  the  preceding  discussion  we  therefore  conclude, 

1.  That  if  an  equation  of  the  second  degree  between  two 
variables  be  resolved  in  respect  to  either  variable,  the  equation 
that  results  from  placing  this  variable  equal  to  that  part  of  its 
value  which  is  independent  of  the  radical  sign  will  be  the  equa 
tion  of  that  diameter  of  the  curve  which  bisects  the  system  of 
chords  parallel  to  the  axis  of  the  variable. 

2.  The  values  of  the  other  variable  found  from  the  equation 
which  results  from  placing  the  guantity  under  the  radical  sign 
equal  to  zero,  in  connection  with  the  corresponding  values  of  the 
first  variable,  will  be  the  co-ordinates  of  the  vertices  of  the 
diameter. 

3.  The  formulas  for  changing  the  reference  of  points 
from  a  system  of  rectangular  co-ordinate  axes  to  any 
other  system  having  a  different  origin  are 

x=a-\-x'coQ.  m-f^/'cos.  n. 
y=b+  z'sin.  m-fy'sin.  n. 


214  ANALYTICAL   GEOMETKY. 

Substituting  these  values  of  x  and  y  in  the  equation 


developing,  and  arranging  the  terms  of  the  resulting 
equation  with  reference  to  the  powers  of  yf  and  xf  and 
their  product,  we  find 

(A  sin.2  n+B  sin.  n  cos.  n+  C  cos.2  ft)  yn 
+(A  sin.2  m+  B  sin.  m  cos.  m-\-  C  cos2  m)  x' 
+\%A  sin.  m  sin.  n+B  (sin.  m  cos.  n 
-fsin  ft  cos.  m)  +2(7  cos.  m  cos.  ft] 


+l(2Ab+Ba+D) 


=0  (1) 


COS.  ft]?/' 

+[2^16+^a+D)  sin. 

cos.  m]x' 

+Abz+Bab+  Ca?+Db+Ea+F. 
Since  we  have  four  arbitrary  quantities,  a,  6,  m,  and  ft 
entering  this  equation  we  may  cause  them  to  satisfy  any 
four  reasonable  conditions.  Let  us  see  if,  by  means  of 
them,  it  be  possible  to  reduce  the  coefficient  of  the  first 
powers,  and  of  the  product  of  the  variables,  separately 
to  zero. 

"We  should  then  have 

f  2A  sin.  m  sin.  n+B  (sin.  m  cos.  ft-fsin.  ftl  __Q 
j         cos.  m)  +2  C  cos.  m  cos.  ft.  j 


sin.  n+(2Ca+Bb+E)  cos.  ft=0        (3) 
sin.  m+(2Ca+Bb+E)  cos.  m=0      (4) 
These  equations  may  be  put  under  the  form 
2J.  tan.  m  tan.  n+B  (tan.  m+tan.  ft)+2<7=0          (20 
(2^6+^a+D)  tan.  n+SGH-^+JE^O  (3/) 

(2Ab+Ba+D)  tan.  m+2(7a+J56+^=0  W 

Now,  since  it  is  necessary  that  m  and  ft  should  differ  in 
value,  it  is  evident  that,  in  order  to  satisfy  eqs.  (3')  and 
(4'),  we  must  have 

2Ab+Ba+D=0  (5) 

And  2Ca+Bb+E=Q  (6) 


INTEKFKETATION   OF   EQUATIONS.     215 

Whence  aJ^AK-BD 

W-A.AG 

And  b 


These  values  of  a  and  b  are  infinite  when  B2 — 41(7=0, 
and  it  will  then  be  impossible  to  satisfy  both  eqs.  (3')  and 
(4'),  and  consequently  to  destroy  the  co-efficients  of  the 
first  powers  of  the  two  variables  in  eq.  (1) ;  we  shall,  for 
the  present,  assume  that  B2 — 41  (7  is  either  greater  or 
less  than  zero. 

By  transposition  and  division  eqs.  (5)  and  (6)  become 

,         B        D 

o = — a — 

21       21 

f)  IT1 
J3  i Mi 

~~~2C      2C 

the  first  of  which,  if  a  and  b  be  regarded  as  variables,  is 
the  equation  of  the  diameter  that  bisects  the  chords  of 
the  curve  which  are  parallel  to  the  axis  of  ?/,  and  the  sec 
ond,  that  of  the  diameter  which  bisects  the  chords  which 
are  parallel  to  the  axis  of  X.  The  values  of  a  and  b, 
given  above,  are,  therefore,  the  co-ordinates  of  the  inter 
section  of  these  diameters. 

Since  eq.  (2')  contains  both  of  the  undetermined  quan 
tities,  m  and  ft,  we  are  at  liberty  to  assume  the  value  of 
either,  and  the  equation  will  then  give  the  value  of  the 
other.  Let  us  take  for  the  new  "axis  of  X  the  diameter 
whose  equation  is 


2A      2A 

T> 

then  tan.  ra= — This  value  of  tan.  m  substituted  in 

2A 
eq.  (2')  gives 

2A(B— B)  tan.  n=52— 41(7, 

Or  l  tan. 


216  ANALYTICAL    GEOMETRY. 

That  is,  the  new  axis  of  y  is  at  right  angles  to  the 
primitive  axis  of  X. 

The  values  of  a,  6,  and  tan.  n  which  we  have  thus 
found,  in  connection  with  the  assumed  value  of  tan.  m, 
will  reduce  the  co-efficients  of  the  first  powers  and  of  the 
product  of  the  variables  in  -eq.  (1)  to  zero. 

To  find  what  the  co-efficients  of  y'2  and  xf2  become,  we 
must  first  get  the  values  of  the  sines  and  cosines  of  the 
angles  m  and  n  from  the  values  of  tan.  m  and  tan.  n. 

T> 

Since  tan.  m= — ,  and  71=90°  we  have 

2JL 

sin.  m=db cos.  m=qc 


sin.  n=~L    cos.  n=Q. 

The  sign  ±  is  written  before  the  value  of  sin.  w,  and 
the  sign  rp  before  that  of  cos.  m,  because  if  the  essential 
sign  of  tan.  m  is  minus,  which  will  be  the  case  when  A 
and  B  have  the  same  sign,  sin.  m  and  cos.  m  must  have 
opposite  signs  ;  but  if  the  essential  sign  of  tan.  m  is  plus, 
then  A  and  B  have  opposite  signs,  and  sin.  m  and  cos.  m 
must  have  like  signs. 

Making  these  substitutions  in  eq.  (1)  it  will  become, 
whether  the  signs  of  A  and  B  are  like  or  unlike, 


Ay'2—  A       —         \  x'2=—  (Ab2+Bab+  Ca2+Db+Ea 

>    4^1  ~"J-JL>      / 

+F.  (!') 

E"ow,  since  the  first  term  of  the  general  equation  may 
always  be  supposed  positive,  the  two  terms  in  the  first 
member  of  equation  (I/)  will  have  like  signs  when  B2  — 
4JLC<0,  and  unlike  signs  when  B2  —  4J.(7>0.  In  the 
first  case  the  form  of  the  equation  is  that  of  the  equation 
of  the  ellipse,  and  in  the  second,  the  form  is  that  of  the 
equation  of  the  hyperbola,  referred  in  either  case,  to  the 
center  and  conjugate  diameters. 


INTERPRETATI     N    OF  EQUATIONS.    217 

Hence,  when  the  transformation  by  which  eq.  (I/)  was 
derived  from  the  general  equation 


is  possible,  we  conclude  that  the   latter  equation  will 
represent  either  the  ellipse,  or  hyperbola,  according  as 


4.  —  Let   us   now   examine  the   case  in  which 
B2—  4-40=0. 

Since,  under  this  hypothesis,  the  co-efficients  of  the 
first  powers  of  both  variables  in  eq.  (1)  cannot  be  de 
stroyed,  we  will  see  if  it  be  possible  to  destroy  the  abso 
lute  term  of  the  equation,  and  the  co-efficients  of  the 
product  of  the  variables,  the  second  power  of  one  varia 
ble  and  the  first  power  of  the  other  variable. 

Then  the  equations  to  be  satisfied  are 


f  2J.sin.wsin.7i-f.B(sin.mcos.tt-fsin.ncos.w) 
\  -f-2(7cos.mcos.n 

A&\n.2m-\-B  sin.m  cos.m-f  <7cos.2m=0.         (8) 
(2Ab+Ba+D)aiu.n+(2Ga+Bb+I!)coB.n==Q.    (3) 
when  it  is  required  that  the  co-efficients  of  x'*  and  yf 
should  reduce  to  zero  in  connection  with  the  absolute 
term  and  the  co-officient  of  x'y',  in  eq.  (1).     To  reduce 
the  co-efficients  of  yfz  and  x'  to  zero,  instead  of  those  of 
x'2  and  y',  it  would  be  necessary  to  replace  eqs.  (8)  and 
(3)  by 

A  sin.2ft-f  B  sin.n  cos.n+  (7cos.2n=0.  (9) 


Equations  (2)  and  (8)  may  be  written 

Atan.2m+B  tan.m-f  (7=0.  (8') 

From  eq.  (8')  we  find 

B       1     |~~  B 

18 


218  ANALYTICAL    GEOMETKY. 

and  this  value  of  tan.  m  substituted  in  eq.  (2')  gives 


or  tan.  n=—. 

73 

That  is,  when  tan.  m  is  equal  to  —  —  ,  eq.   (2')  and, 

therefore,  eq.  (2),  will  be  satisfied  independently  of  the 
angle  n. 

Equation  (7),  being  what  the  general  equation  becomes 
when  a  and  b  take  the  place  of  x  and  y  respectively, 
shows  that  the  new  origin  of  co-ordinates  must  be  on  the 
curve.  Solving  this  equation  with  reference  to  6,  and 
introducing  the  condition  B2  —  4J.(7=0,  we  find 


Now,  because  the  imposed  conditions  require  that  the 
transformed  equation  shall  be  of  the  form 


it  follows  that  every  value  of  x*  must  give  two  numeri 
cally  equal  values  of  y'  ;  hence,  the  new  axis  of  Y  must 
be  parallel  to  the  system  of  chords  bisected  by  the  new 
axis  of  X.  That  is,  n  must  be  equal  to  90°,  and,  conse 
quently,  sin.ft=l,  cos.  7i=0. 

Equation  (3)  will  therefore  become 


Whence  b=  —  _  a  —  _  ,  and  the  radical  part  of  the 

2A      2A' 

value  of  b  will  disappear,  or  we  shall  have 
2(_RD— 

From  which  we  get 

= 

These  values  of  a  and  b  place  the  new  origin  at  the 
vertex  of  the  diameter  whose  equation  is 

y  --  ^-^, 

2A      2A' 


INTERPRETATION   OF   EQUATIONS.     219 

and  make  the  new  axis  of  Y  a  tangent  line  to  the  curve 
at  the  vertex  of  this  diameter. 

The  values  of  a,  6,  m  and  n  which  we  have  now  found, 
substituted  in  eq.  (1),  will  reduce  it  to 


Or 


JO. 

Denoting  the  co-efficient  of  xf  by  —  2p',  this  last  equa 
tion  becomes 

y'3=2p'z',  (10) 

which  is  of  the  form  of  the  equation  of  the  parabola  re 
ferred  to  a  tangent  line  and  the  diameter  passing  through 
the  point  of  contact. 

The  transformation  by  which  eq.  (10)  was  derived  from 
the  general  equation  is  always  possible  when  JS2  —  4AC 
=0,  unless  we  also  have  BD  —  %AE=Q.  If  we  suppose 
that  both  of  these  conditions  are  satisfied,  the  general 
value  of  y,  which  is 


reduces  to 


whence 


and 


which  are  the  equations  of  two  parallel  straight  lines. 

Under  the  suppositions  just  made,  the  general  equa 
tion  may  be  written  under  the  form 


which  may  be  satisfied  by  making,  first  one,  then  the 
other  factor  of  the  first  member,  equal  to  zero.     Each  of 


220  ANALYTICAL   GEOMETRY. 

the  equations  thus  obtained,  being  of  the  first  degree  in 
respect  to  x  and  y,  will  represent  a  right  line. 

If  the  further  condition,  D2  —  4J.J7r<0,  be  imposed,  the 
right  lines  will  have  no  existence,  and  the  general  equa 
tion  can  be  satisfied  by  no  real  values  of  x  and  y. 

The  value  of  2p',  the  parameter  of  the  diameter  which 
becomes  the  new  axis  of  JT,  will  be  found  by  substituting 
in  the  expression 


the  values  of  a,  b  and  cos.  m.     These  values  ajre 


cos.  m= 

To  reduce  eq.  (1)  to  the  form 

z'2=2/y  (11) 

we  must  satisfy  equations  (7),  (2),  (9)  and  (4). 

T> 

From  eq.  (9)  we  find  tan.  n—  —  —  -,  and  this  value  of 

2iA 

tan.  n  substituted  in  eq.   (2r)   gives  tan.   w=-^>  resu^s 

which  might  have  been  anticipated,  since  eqs.  (3)  and  (4) 
are  the  same,  except  that  m  in  the  former  takes  the  place 
of  n  in  the  latter. 

Because  eq.  (11)  will  give  two  numerically  equal  val 
ues  of  x'  for  every  value  of  yf  ,  the  new  axis  of  JTmust 
be  parallel  to  the  system  of  chords  bisected  by  the  new 
axis  of  Y;  hence  m=0°,  sin.  w=0,  cos.  w=l,  and  equa 
tion  (4)  therefore  reduces  to 


-r>  ~ffl 

Whence  a=  —  __  b  —  _     - 

2(7       2(7 

Solving  eq.  (7)  with  reference  to  a  we  have 


INTERPRETATION    OF  EQUATIONS.    221 

T)  ~fjl  -I 


a= 


By  comparing  this  value  of  a  with  that  which  precedes 
we  find 


Whence  b=_E*-4CF 


These  values  of  a  and  6  place  the  new  origin  at  the 
vertex  of  the  diameter  whose  equation  is 

*—£*-£. 

20      2(7 

,-?!»-! 

The  transformation  by  which  eq.  (4)  is  derived  from 
eq.  (1)  will  be  impossible  when  b  is  infinite  ;  that  is  when 


It  may  be  easily  proved  that  when  W  —  4  A  (7=0,  the 
condition  BD  —  2AE=Q  necessarily  includes  the  condi 
tion  BE  —  2(7.D=0  ;  that  is,  when  we  cannot  transform 
eq.  (1)  into  eq.  (10),  it  will  also  be  impossible  to  trans 
form  it  into  eq.  (11). 

For  BD—  2AJE=Q 


And  ^—4^(7=0  gives     j 

2(7    E 
Whence  ""=  °>  or 


5.  —  "We  have  now  established  the  following  criteria  for 
the  interpretation  of  any  equation  of  the  second  degree 
between  two  variables,  viz  : 

For  the  ellipse,  B2—  4AC<0. 
For  the  hyperbola,  B2—  4AC>0. 
For  the  parabola,  B2—  4J.G=0. 

It  remains  for  us  to  indicate  the  construction  of  any 
of  these  curves  from  its  equation,  and  in  doing  this,  we 
19* 


222  ANALYTICAL    GEOMETRY. 

shall  follow  the  order  in  which  the  conditions  are  given 
above. 

First,  B*—  4AC<0,  the  ellipse. 

6. — Let  us  resume  the"  formulas. 
aJlAE—BD 


B*-4AC  2 A 

— A  (B2—^AC\r'2=—(A  b  *+Bab+  Ca 


and  suppose,  for  a  particular  case,  J3=0,  and  A=C. 

E    i         D 

"We  shall  then  have  a= — ;r-p  b HI 


And  yf2+xf2 

That  is,  the  general  equation,  under  the  suppositions 

made,  represents  a  circle  having  a= — r-j>  ^= — ir-4^or^e 

&-A-  Z-A 

co-ordinates  of  its  center,  and   I        '  ^AU  fOT  ^s  ra, 

\  4A2 

dius. 

Draw  AX^  A  Y  for  the  primitive 
co-ordinate    axes,  lay    off   AB— 

77T  T\ 

— , AD= — - — ,  and  through  the 

points  B  and  D  draw  the  parallels 
jB(7and  DC  to  the  axes.  Their 
intersection,  (7,  is  the  center  of  the 
circle,  and  the  circumference  de 
scribed  with  CE=  l-Lr+H'  £A  F  ag  a  ra^iug  win  "be 

N  4A2 

that  represented  by  the  given  equation. 

The  general  equation  gives 


INTERPRETATION   OF   EQUATIONS.     223 
B        D       1 


Placing  the  quantity  under  the  radical  sign,  in  this 
value  of  y,  equal  to  zero,  we  have 


and  denoting  the  roots  of  this  equation  by  xf  and  z",  the 
value  of  y  may  be  written 


Now  x'  and  x"  are  the  abscissas  of  the  vertices  of  the 
diameter  whose  equation  is 


2A       2A 

The  corresponding  values  of  y  are 
f==_Bx'+D 

Bx"+D 


2A 

Substituting  these  values  of  #',  x"  and  #',  y"  in  the  for 
mula 


we  have  x     x    \B2-\-4A2  ^or  ^ne  length  of  the  diameter. 

The  diameter  which  is  conjugate  to  this  is  that  which  is 
parallel  to  the  axis  of  y.     We  find  the  ordinates  of  its 

xf+x" 

vertices  by  substituting  a=  —  -  —  for  x  in  eq.  (q),  which 

2 


then  becomes 


B(x'+x"}     Z>   ,x'— 


y=      ~^A~   "lA-lA-^0-^ 

Tb-oL  &-c\-  rt-n.      ^^ 

Denoting  these  two  values  of  y  by  y  ^  y^  their  differ 
ence,  which  is  the  length  of  the  conjugate  diameter,  is 

x'— x" 


224 


ANALYTICAL    GEOMETRY. 


To  find  the  angle  that  the  con 
jugate  diameters  make  with  each 
other,  let  VVr  be  the  first  diameter 
and  QQ'  the  second.  The  angle 
that  W  makes  with  the  axis  of 
X  is  equal  to  V  VR,  and  its  cosine 


B     D     E 


VE 


vv> 


x 


s 


and  the  [_QCVf=t^Q  t_BVVf=$Q°+the  \_V'VE. 

"When  the  roots  of  eq.  (p)  are  equal,  the  vertices  of  the 
first  diameter,  and  also  those  of  its  conjugate,  coincide, 
and  the  ellipse  reduces  to  a  point.  Equation  (q)  may 
then  be  put  under  the  form 


Because  JB2  —  &AC  is  negative,  this  value  of  y  will  be 
imaginary  for  every  value  of  x  except  the  particular  one, 
x=xr,  which  causes  the  radical  to  disappear. 

When  the  roots  of  eq.  (p)  are  real  and  unequal,  that 
one  of  the  factors  (x  —  x'\  (x  —  x")  under  the  radical  in  eq. 
(q),  which  corresponds  to  the  root  which  is  algebraically 
the  greater,  will  be  negative,  while  the  other  will  be  pos 
itive,  for  all  values  of  x  included  between  the  limits  of 
the  smaller  and  greater  roots.  The  quantity  under  the 
radical,  being  then  composed  of  the  product  of  three 
factors,  two  of  which  are  negative  and  one  positive,  will 
itself  be  positive  and  the  corresponding  values  of  y  will 
therefore  be  real. 

All  values  of  x  which  exceed  the  greater,  and,  also,  all 
values  of  x  which  are  less  than  the  smaller,  of  these  roots, 
will  render  the  quantity  under  the  radical  negative  and 
the  corresponding  values  of  ?/  imaginary.  The  roots  x' 
and  x"  are  therefore  the  limits  within  which  we  would 


INTERPRETATION  OF  EQUATIONS.     225 

select  values  of  x  to  substitute  in  the  equation  to  get  the 
co-ordinates  of  points  of  the  curve. 

When  the  roots  of  eq.  (p)  are  imaginary,  the  product 
of  the  factors  (x  —  x'\  (x  —  x"]  under  the  radical  in  eq.  (q) 
will  remain  positive  for  all  real  values  of  x;  and  because 
the  other  factor  is  IP  —  4:A  (7<0,  the  radical  will  always 
be  imaginary  :  that  is,  no  real  value  of  x  which  will  give 
a  real  value  for  y.  There  is,  then,  in  this  case,  no  point 
in  the  plane  of  the  co-ordinate  axes  whose  co-ordinates 
will  satisfy  eq.  (q),  and,  consequently,  the  equation  from 
which  it  was  derived,  and  the  curve,  has  no  existence,  or 
it  is  imaginary. 

By  the  solution  of  eq.  (p)  it  will  be  found  that  when 
the  expression 


is  positive,  the  roots  of  the  equation  are  real  and  unequal  ; 
when  the  expression  is  zero  the  roots  are  real  and  equal, 
and  when  negative  the  roots  are  imaginary. 

If  we  solve  the  general  equation  with  reference  to  x 
instead  of  y,  and  place  the  quantity  under  the  radical 
sign  equal  to  zero,  we  shall  find  that  when  the  expression 

(BE—Z  CD)2—(£2—4AC)  (E2—±  CF) 
is  positive,  the  roots  of  the  resulting  equation  are  real 
and  unequal  ;  when  zero,  these  roots  are  real  and  equal, 
and  when  negative  they  are  imaginary. 

It  might  be  inferred  that  if  these  roots  are  real  and 
unequal,  equal,  or  imaginary  when  the  general  equation 
is  resolved  with  reference  to  one  variable,  they  would  be 
like  characterized  when  it  is  resolved  with  reference  to 
the  other.  To  prove  this,  we  develope  the  first  of  the 
above  expressions  and  find  that  it  becomes 

4J.  (A(fi)*+  C(D}2+F(B)2—BDE-4A  OF.) 
The  development  of  the  second  is 


226 


ANALYTICAL    GEOMETRY. 


+  C(D}z+F(Bf—BDE-4ACF.\ 

The  only  difference  in  these  developments  is  that  the 
coefficient  of  the  parenthesis  in  the  first  is  4A,  and  in  the 
second  it  is  4(7;  but  when  jB2  —  4J.(7<0,  A  and  C  must 
have  the  same  sign,  hence  these  expressions  must  be  posi 
tive,  negative,  or  zero  at  the  same  time. 

Second,  B2  —  4JL(7>0,  the  hyperbola. 

7.  —  "We  will  begin  by  supposing  B=Q,  and  A—  —  C. 

The  formulas  for  a,  b  and  tan.  m  will  then  give 

Tjl  J~\ 

a—  —  ,  b=  —  _  .  tan.ra=0, 
2A*  2A' 

and  eq,  (!')  will  become 


4J. 


This  is  the  equation  of  an  equilateral  hyperbola  whose 
semi-axis  is  the  square  root  of  the  numerical  value  of  the 

expression .     Since  tan.  m=0,  m=0,  and 

one  of  the  axes  of  the  hyperbola  is  parallel  and  the  other 
perpendicular  to  the  primitive  axis  of  X.     If  the  sign  of 

iF 

-   is  negative,  the  transverse  is  the  parallel 


axis ;  if  negative,  it  is  the  perpendicular  axis. 

To  construct  the   curve,  let  AX 
and  A  Fbe  the  primitive  co-ordinate 

axes.     Lay  off  the  positive  abscissa 
jji 

==n—.')  and  the  negative  ordinate 
ZuoL 

= — —  ;  the  parallels  to  the  axes 
2A 

drawn  through  D  and  E  will  be  the  axes  of  the  hyper 
bola,  and  C  will  be  its  center.  On  these  axes,  lay  off 
from  the  center,  the  distances  CV,  CV>  CR,  CR',  each 


INTERPRETATION    OF  EQUATIONS.    227 


equal  to 

\ 


P—IP—  4  AF  and  we  have  ^e  ^eQ  of  con_ 
~     ~~ 


jugate  equilateral  hyperbolas.  The  foci  may  be  found 
by  describing  a  circumference  with  C  as  a  center  and  CH, 
the  hypothenuse  of  the  isosceles  right-angled  triangle 
CVH,  as  a  radius  ;  the  circumference  will  intersect  the 
axes  at  the  foci. 

For  another  case,  let  us  suppose  -4=0  and  C=0  ;  then 

T) 

the  value  —  —-which  was  assumed  for  tan.  m  becomes 
2J. 

infinite,  or  the  new  axis  of  X  is  perpendicular  to  the 
primitive  axis  of  X,  and  since  tan.  n  is  also  infinite,  the 
new  co-ordinates  axes  would  coincide  ;  in  other  words, 
with  this  value  of  tan.  m,  it  would  be  impossible,  under 
the  hypothesis,  to  transform  the  original  equation  into 
eq.  (V}.  But  if  J.=0,  and  (7=0,  the  co-efficient  of  x'y' 
in  eq.  (1)  becomes 

-B(sin.  m  cos.  71+  sin.  n  cos.  m). 

Placing  this  equal  to  zero,  and  dividing  through  by 
B  cos.  m  cos.  7i,  we  have 

tan.  m-f  tan.  7i=0, 

Or  tan.  m=  —  tan.  n. 

Since  we  are  at  liberty  to  select  a  value  for  either  m  or 
7i,  let  us  make  71=45°  ;  then  m=  —  45°.  The  values  of  a 
and  6,  which  will  destroy  the  co-efficients  of  xr  and  yf 

D  ,        E 
a=-_,6=-_. 

Substituting  these  values  in  eq.  (1),  reducing  and  trans 
posing,  we  have 


which  is  also  the  equation  of  the  equilateral  hyperbola, 

D  E 

the  co-ordinates  of  whose  center  are  a=  —  ^,  6  --  -„> 

JD  JL> 


228  ANALYTICAL   GEOMETRY. 

and  whose  semi-axis  is  the  square  root  of  the  numerical 

value  of  -i  -  —  --  L   The  asymptotes  of  this  hyperbola 

O/  7~)  TT       7-?  77^ 

are  parallel  to  the  primitive  axes,  and  if   3  —  _  ---  is 

negative,  the  transverse  axis  makes  a  negative  angle  with 
the  primitive  axis  of  JT,  if  positive,  it  makes  a  positive 
angle  with  that  axis. 

There  is  another  case  in  which  the  transformation  by 
which  eq.   (V)  was  obtained,  cannot  be  made  with  the 

73 

value  —  —  for  tan  m.     It  is  that  in  which  A  becomes  zero, 
2-<4_ 

and  C  does  not.  We  then  assume  for  tan.  m  the  tangent 
of  the  angle  that  the  diameter  whose  equation  is 

B    _E 
~~Wy    20 
makes  with  the  axis  of  X.     That  is,  we  make 

2  G 
tan.  m=  —  — 

75 

Proceeding  with  this  as  with  the  value  —  —  ,    we    shall 

2A 

find  for  the  transformed  equation 


By  making  A=Q,  this  equation  becomes 


which  is  that  of  an  hyperbola  referred  to  a  system  of 
conjugate  diameters,  one  of  which  bisects  the  chords 
which  are  parallel  to  the  primitive  axis  of  X. 

In  the  general  case  the  course  to  be  pursued  for  the 
hyperbola  differs  so  little  from  that  already  indicated  for 
the  ellipse,  that  it  is  unnecessary  to  dwell  upon  it  at 
length. 


INTERPRETATION   OF   EQUATIONS.     229 

The  quantity  under  the  radical  in  the  general  value 
of  y  placed  equal  to  zero  gives  the  equation 

> 

The  roots  of  this  equation  are  the  abscissas  of  the  ver 
tices  of  the  diameter,  whose  equation  is 

y=-^x-°. 

2A      2A 

When  these  roots  are  real  and  unequal,  the  diameter 
terminates  in  the  hyperbola;  when  imaginary,  it  termi 
nates  in  the  conjugate  hyperbola. 

Denoting  these  abscissas,  when  real,  by  xf  and  x",  and 
the  corresponding  ordinates  by  y'  and  y,  we  have 

Bx'+D 


y1- 
f- 


2A 

Bx"+D 


2A 
By  placing  these  values  of  a/,  x"  and  y',  y"  in  the  for 


mula 


we  shall  have  the  length  of  the  diameter,  and  the  angle 
included  between  it  and  its  conjugate  will  be  found  pre 
cisely  as  in  the  ellipse. 

If  xf  be  the  smaller  and  x"  the  greater  abscissa,  then  all 
values  of  x  between  x'  and  x"  will  give  imaginary  values 
for  ?/,  and  will  answer  to  no  points  of  the  curve ;  but  all 
values  of  x  less  than  x1 ',  and  also  all  values  of  x  greater 
than  x"  will  give  real  values  for  #',  and  such  values  of  x 
with  the  corresponding  values  of  y  will  be  the  co-ordi 
nates  of  points  of  the  hyperbola. 

When  the  roots  x' ,  x"  are  imaginary,  the  diameter 
whose  equation  is 


2A       2A 
20 


230  ANALYTICAL    GEOMETRY. 

terminates  in  the  hyperbola  which  is  conjugated  to  that 
represented  by  the  given  equation,  and  the  diameter 
which  is  conjugate  to  this  diameter  will  terminate  in  the 
given  hyperbola. 

The  conjugate  diameter  may  be  found  in  the  case  of 
both  the  ellipse  and  hyperbola  by  making  first  ?/'=0  in 
eq.  (I'),  and  taking  the  square  root  of  the  corresponding 
numerical  value  of  x'2,  and  then  £'=(),  and  taking  the 
square  root  of  the  corresponding  numerical  value  of  yn. 

8.  —  In  the  transformation  of  co-ordinates  by  which  the 
original  equation  was  changed  into  eq.  (1)  had  the  condi 
tion,  that  the  new  co-ordinate  axes  should  be  rectangular, 
been  imposed,  as  it  might,  we  would  have  had  n  —  m=90°, 
n=  90°  +m.  Sin.  71=  cos.  m,  cos.  n—  —  sin.  m. 

These  values  being  substituted  in  eq.  (2)  will  give 
2A  sin.  m  cos.  m  —  B  sin.2m-fJ3cos.2m  —  2(7sin.mcos.m=0, 
which,   by   dividing  through  by  cos.2m,  and   denoting 
sin'm  by  *,  becomes 

COS.  ill 


Whence        ^ 


Since  the  product  of  these  two  values  of  t  is  equal  to 
—  1,  they  are  the  tangents  of  the  angles  that  two  straight 
lines  at  right  angles  to  each  other  make  with  the  axis  of 
X.  Now,  if  eqs.  (5)  and  (6)  are  satisfied  at  the  same 
time  ;  that  is,  if  the  new  origin  be  placed  at  the  point  of 
which  the  co-ordinates  are 


the  values  of  t  just  found  will  be  the  tangents  of  the 
angles  that  the  axes  of  the  ellipse,  or  hyperbola,  as  the 
case  may  be,  make  with  the  primitive  axis  of  X.  De 
noting  these  tangents  by  tr  and  t",  we  shall  have 


INTERPRETATION   OF  EQUATIONS.     231 

y—b=t'(x—a), 


for  the  equations  of  the  axes,  and  by  combining  the 
equations  of  the  axes  with  the  original  equation,  we  may 
find  the  co-ordinates  of  their  vertices,  and,  consequently, 
their  length. 

9.  —  When  the  roots  xf  and  x"  become  equal,  the  value 
of  y  may  be  written 


JBx+D,  x—  x 


y=  —  _  __     __    Mo2     4  A  n 
2A         2A  >p 

For  the  hyperbola,  HP  —  4JL(7>0,  and  these  values  of  y 
are  real.     We  therefore  have 


These  equations  represent  two  right  lines,  and,  since 
the  co-efficients  of  x9  when  the  second  members  are  ar 
ranged  with  reference  to  it,  are  different,  these  lines  will 
intersect.  We  see  that  by  making  x=x',  the  two  equa 
tions  will  give  the  same  value  for  y.  Hence,  x=x',  and 

y=  —  are  the  co-ordinates  of  the  intersection  of 

2A 

the  lines. 

The  line  BE,  whose  equation  is 

y=-  *^, 

2A      2A' 

still  has  the  property  of  bisecting  all 
lines  drawn  parallel  to  the  axis  of 
Y",  which  are  limited  by  the  lines 
BC  and  BD,  whose  equations  are  eqs.  (r)  and  (s). 

Third,  B2  —  4L4C==0,  the  parabola. 

10.  —  The  equation  of  the  diameter  that  bisects  the 
chords  of  the  curve  which  are  parallel  to  the  axis  of  Y  is 


232  ANALYTICAL    GEOMETRY. 


and  that  of  the  diameter  which  bisects  the  chords  paral 
lel  to  the  axis  of  JTis 


2C      E 
y=  --^-ff 

Since  a  tangent  line  drawn  through  the  vertex  of  a  di 
ameter  is  parallel  to  the  chords  that  the  diameter  bisects, 
it  follows  that  the  diameters  represented  by  the  above 
equations  are  perpendicular  to  each  other,  and,  therefore, 
(Prop.  5,  Chap.  4),  their  intersection,  in  the  case  of  the 
parabola,  is  on  ftie  directrix. 

The  abscissa  of  the  vertex  of  the  first  diameter  is  the 
value  of  x  given  by  the  equation  , 

2(J3D—  ZAE}x+D*—  4^=0, 

the  first  member  of  which  is  the  quantity  under  the  radical 
in  the  general  value  of  ?/,  after  we  have  made  HP  —  4./i(7=0. 

Denoting  this  abscissa  by  x'  we  have 


If  we  denote  the  co-ordinates  of  the  vertex  of  the  sec 
ond  diameter  by  x"  and  y,  we  have 
,,=        ^2—  4  CF 


2C 

Let  Pand  Pf  be  the  two  vertices  thus  found.  Through 
the  first  draw  PT  parallel  to  the  axis  of  F,  and  through 
the  second,  Pf  T  parallel  to  the  axis  of  X.  These  lines 
will  be  tangent  to  the  parabola  at  P  and  Pr  respectively, 


INTERPRETATION    OF   EQUATIONS.     233 


and  their  intersection,  T,  will  be 
a  point  of  the  directrix.  The 
lines  CM,  BN,  drawn  through 
P  and  P;,  making,  with  the  axis 
of  Jfj  angles  having  for  their 
common  tangent 

_B_     JLC_ 
2A         B* 


\ 


are  diameters  of  the  curve,  and 
BC  drawn  through  T  perpendicular  to  these  diameters, 
is  the  directrix.  With  P  as  a  center  and  PC  as  a  radius, 
or  with  P'  as  a  center  and  P'B  as  a  radius,  describe  an 
arc  of  a  circle.  This  arc  will  cut  the  chord  PPf  at  the 
focus  F.  The  perpendicular  FD,  drawn  through  F  to 
the  directrix,  is  the  axis,  and  the  middle  point,  V,  of  FD, 
is  the  vertex  of  the  parabola. 

EXAMPLES. 

It  will  aid  in  the  construction  of  the  curve  represented 
by  any  equation  to  find  the  points  in  which  it  is  inter 
sected  by  the  co-ordinate  axes.  If  we  make  either  vari 
able  equal  to  zero  in  the  equation,  the  values  of  the  other 
variable  given  by  the  resulting  equation  will  be  the  dis 
tances  from  the  origin  to  the  intersections  of  the  curve, 
with  axis  of  the  latter  variable.  When  the  roots  of  the 
equation  which  we  solve  are  real  and  unequal,  there  will 
be  two  intersections,  where  real  and  equal,  the  axis  will 
be  tangent  to  the  curve  at  the  point  thus  determined,  and 
when  imaginary,  the  curve  and  the  axis  will  have  no 
common  points. 

1. — Construct  the  curve  represented  by  the  equation 


Whence 

Here  J 
20* 


_  2x(x—  2). 
,  JB=2,   (7=3;  therefore  B2—  4AC<0,  and 


234 


ANALYTICAL   GEOMETRY. 


the  curve  is  an  ellipse  which  passes  through  the  origin 
of  co-ordinates,  since  the  equation  has  no  absolute  term. 

y=—x 

is  the  equation  of  a  diameter  of  the 
curve  and  the  co-ordinates  of  its  ver 
tices  are  x'  =0,  yf=  0  and  x"=  2,  /=—  2. 
By  making  x=~L  in  the  original  equa 
tion,  we  find  y=+.  41-f,  or  —  2.41 
for  the  ordinates  of  the  vertices  of  the 
diameter  conjugate  to  the  first. 

The  length  of  the  first  diameter  is 
equal  to   ^8=2.82+,  and  the  length  of  the  second  is 
+.41+2.41=2.82. 

2.  —  Determine  the  curve  that  corresponds  to  the  equation 

y*+2xy+x2—  6y-f9=0. 

Here^=l,  .£=2,  (7=1,  hence  B2—  4^(7=0,  and  the 
curve  is  a  parabola.     We  find 


And  z=—  y±§y—  9. 

The  diameter  whose  equation  is  y=  —  £+3  has  x'=0, 
and  ?/'=3  for  the  co-ordinates  of  its  vertex.  The  axis  of 
y  is  therefore  tangent  to  the  curve.  The  co-ordinates  of 
the  vertex  of  the  diameter  whose  equation  is  x=  —  y  are, 
x"=  —  1J,  and  y=l|,  and  a  line  drawn  through  this  point 
parallel  to  the  axis  of  X  will  be  tangent  to  the  curve. 

Let  P'  be  the  vertex  of  the  first 
diameter  and  P  that  of  the  second. 
The  chord  PP'  passes  through  the 
focus.  P'S',  PS  making  with  the 
axis  of  X,  on  the  negative  side? 
angles  of  45°  are  diameters  of  the 
curve,  and  B  T  a  perpendicular  to 
P$is  the  directrix. 


X 


INTERPRETATION   OF   EQUATIONS.     235 

3. — Determine  the  curve  of  which  the  equation  is 

In  this  case  A=l,  5=2,   C=— 2 ;  hence  W— 4AC>0, 
and  the  curve  is  an  hyperbola.     The  equation  gives 

The  abscissas  of  the  vertices  of  the  diameter  whose 
equation  is 

2/=—  x+2 
are  the  roots  of  the  equation 

Whence  xf= — 1,  and  x"=2,  and  the  corresponding  val 
ues  of  y  are  y'=S  and  ?/"=0. 

The  diameter  which  is  parallel  to 
the  axis  of  y  is  conjugate  to  PP\ 
and  terminates  in  the  conjugate  hy 
perbola.  The  co-ordinates  of  its 
vertices  are  imaginary  and  may  be 
found  by  making  x=-|  in  the  original 
equation.  We  would  thus  find 

2 

The  conjugate  diameter  will  therefore  be  about  5.2. 
The  point  E  in  which  the  curve  intersects  the  axis  of  X 
is  on  the  left  of  the  origin  and  at  a  distance  from  it  equal 
to  2J  units. 

4. — Determine  the  curve  represented  by  the  equation 

In  this,  the  condition  B2 — 4AC=Q  is  satisfied,  and  the 
curve  is  the  parabola  ;  but  it  answers  to  the  case  in  which 
the  parabola  reduces  to  two  parallel  lines. 

In  fact  the  equation  may  be  put  under  the  form 


Whence 

Or  2/4-3z=5  or  —3. 


236  ANALYTICAL    GEOMETRY. 

The  first  member  of  the  equation  may  therefore  be  re 
solved  into  the  factors  y+Sx  —  5,  and  y+3x+&,  which, 
placed  separately  equal  to  zero,  give  for  the  parallel  lines 
the  equations 


And  y=  —  3x  —  3. 

5.  —  Determine  the  curve  of  which  the  equation  is 


In  this  we  have  -B2  —  4  A  (7<0,  and  the  curve  is  an  ellipse, 
but  it  answers  to  the  case  in  which  the  curve  becomes 
imaginary.  For,  resolving  the  equation  in  relation  to  y, 
we  find 


(x—  2)2. 

The  quantity  under  the  radical  in  this  value  of  y  will 
be  negative  for  every  real  value  of  cc,  hence,  al]  values  of 
y  are  imaginary  ;  that  is,  there  is  no  point  whose  co-ordi 
nates  will  satisfy  the  given  equation. 

By  inspection  we  may  also  discover  that  the  first  mem 
ber  of  the  equation  can  be  placed  under  the  form 
(2/_2z-l)2+(r-2)2, 

which  is  the  sum  of  two  squares,  and  must  therefore  re 
main  positive  for  all  real  values  of  x  and  y. 

6.  —  What  kind  of  a  curve  corresponds  to  the  equation 


Ans.  It  is  an  hyperbola.  The  axis  of  Y  is  midway  be 
tween  the  two  branches.  One  branch  of  the  curve  cuts 
the  axis  of  X  at  the  point  —  1  ;  the  other  branch  cuts  the 
same  axis  at  the  point  -f  3. 

7.—  Determine  the  curve  represented  by  the  equation 


Eesolving,  we  find 

(y—x)2+(x—  1)2+3=0. 


INTERSECTION  OF  LINES.  237 

The  condition  for  the  ellipse  is  satisfied,  but  the  curve 
is  imaginary. 

8.  —  What  kind  of  a  curve  corresponds  to  the  equation 


Ans.  It  is  a  parabola  passing  through  the  origin  and  ex 
tending  without  limit,  in  the  direction  of  x  and  y  negative. 

9.  —  What  kind  of  a  curve  corresponds  to  the  equation 


Ans.  It  is  a  parabola,  cutting  the  axis  of  X  at  the  dis 
tance  of  —  1  and  -f  1  from  the  origin,  and  extending  in 
definitely  in  the  direction  of  plus  x  and  plus  y. 

10.  —  What  kind  of  a  curve  corresponds  to  the  equation 


Ans.  It  is  a  straight  line  passing  through  the  origin, 
making  an  angle  of  26°  34'  with  the  axis  of  Y. 

11.  —  What  kind  of  a  curve  corresponds  to  the  equation 


Ans.  It  is  an  ellipse  limited  by  parallels  to  the  axis  of 
Y  drawn  through  the  points  —  1,  and  +1,  on  the  axis 
of  X. 

CHAPTER  VII. 

ON   THE   INTERSECTIONS    OF  LINES   AND  THE    GEOME 
TRICAL  SOLUTION  OF  EQUATIONS. 

"We  have  seen  that  the  equation  of  a  straight  line  is 

y—tx-\-c^ 
And  that  the  general  equation  of  a  circle  is 

(x±a)*+(y±b)*=E2. 
The  first  is  a  simple,  the  second  a  quadratic  equation, 


238  ANALYTICAL    GEOMETRY. 

and  if  the  value  of  x  derived  from  the  first  be  substituted 
in  the  second,  we  shall  have  a  resulting  equation  of  the 
second  degree,  in  which  y  cannot  correspond  to  every 
point  in  the  straight  line,  nor  to  every  point  in  the  cir 
cumference  of  the  circle,  but  it  will  correspond  to  the  two 
points  in  which  the  straight  line  cuts  the  circumference, 
and  to  those  points  only. 

And  if  the  straight  line  should  not  cut  the  circumfer 
ence,  the  values  of  y  in  the  resulting  equation  must  neces 
sarily  become  imaginary.  All  this  has  been  shown  in  the 
application  of  the  polar  equation  of  the  circle,  in  Chap.  2. 

Let  us  now  extend  this  principle  still  further.  The 
equation  of  the  parabola  is 


an  equation  of  the  second  degree,  and  the  equation  of  a 
circle  is 


also  an  equation  of  the  second  degree.  But  when  two 
equations  of  the  second  degree  are  combined,  they  will 
produce  an  equation  of  the  fourth  degree. 

But  this  resulting  equation  of  the  fourth  degree  can 
not  correspond  to  all  points  in  the  parabola,  nor  to  all 
points  in  the  circumference  of  the  circle,  but  it  must  cor 
respond  equally  to  both  ;  hence,  it  will  correspond  to  the 
points  of  intersection,  and  if  the  two  curves  do  not  in 
tersect,  the  combination  of  their  equations  will  produce 
an  equation  whose  roots  are  imaginary. 

Let  us  take  the  equation  y2=2px,  and  take  p  for  the 
unit  of  measure,  (that  is,  the  distance  from  the  directrix 

to  the  focus  is  unity,)  then  z=^_,  and  this  value  of  x 

2t 
substituted  in  the  equation  of  the  circle,  will  give 


INTERSECTION   OF    LINES.  239 

Let  the  vertex  of  the  parabola     Y 
be  the  origin    of  rectangular  co 
ordinates. 

Take  AP=x,  and  let  it  refer  to 
either  the  parabola  or  the  circle, 
and  let  PM=y,  AF=^  AH=a, 
JHC=b,  and  CM=E. 

ISTow  in  the  right  angle  triangle  */ 

CMD,  we  have 


and  corresponding  to  this  particular  figure,  we  shall  have 
in  lieu  of  the  preceding  equation 


"Whence    ^/4+(4—  4%2—  8%=4(^2—  a2—  b\)    (F) 
This  equation  is  of  the  fourth  degree,  hence  it  must 
have  four  roots,  and  this  corresponds  with  the  figure,  for 
the  circle  cuts  the  parabola  in  four  points,  M,  Mf,  M", 
and  M"r. 

The  second  term  of  the  equation  is  wanting,  that  is, 
the  co-efficient  to  y*  is  0,  and  hence  it  follows  from  the 
theory  of  equations,  that  the  sum  of  the  four  roots  must 
be  zero. 

The  sum  of  two  of  them,  which  are  above  the  axis  of 
A  X,  (the  two  plus  roots,)  must  be  equal  to  the  sum  of 
the  two  minus  roots  corresponding  to  the  points  M" 
and  M  '". 

The  values  of  a  and  b  and  R  may  be  such  as  to  place 
the  center  C  in  such  a  position  that  the  circumference  can 
cut  the  parabola  in  only  two  points,  and  then  the  result 
ing  equation  will  be  such  as  to  give  two  real  and  two 
imaginary  roots. 

Indeed,  a  circumference  referred  to  the  same  unit  of 
measure  and  to  the  same  co-ordinates,  might  not  cut  the 


240  ANALYTICAL    GEOMETRY. 

parabola  at  all,  and  in  that  case  the  resulting  equation 
would  have  only  imaginary  roots. 

In  case  the  circle  touches  the  parabola,  the  equation  will  have 
two  equal  roots. 

Now  it  is  plain  that  if  we  can  construct  a  figure  that  will 
truly  represent  any  equation  in  this  form,  that  figure  will  be  a 
solution  to  the  equation.  For  instance,  a  figure  correctly 
drawn  will  show  the  magnitude  of  PM,  one  of  the  roots 
of  the  equation. 

We  will  illustrate  by  the  following 

EXAMPLES. 
1.  —  Find  the  roots  of  the  equation 

y_-11.14?/2—  6.74^+9.9225=0. 

This  equation  is  the  same  in  form  as  our  theoretical 
equation  (F),  and  therefore  we  can  solve  it  geometrically  as 
follows  : 

Draw  rectangular  co-ordinates,  as  in  the  figure,  and 
take  AF=^  and  construct  the  parabola. 

To  find  the  center  of  the  circle  and  the  radius,  we  put 

4—  4a=—  11.14,         (1)          —86=—  6.74,         (2-) 
and  4(^2—  a2—  62)=—  9.9225.  (3) 

From  eq.  (1),  a=3.78.     From  eq.  (2),  6=0.84. 
And  these  values  of  a  and  6,  substituted  in  eq.  (3),  give 
.£=3.34,  nearly. 

Take  from  the  'scale  which  cor-    y 
responds  to  AF=±,  ^LZT=a=3.78, 
HC=  0.  84,  and  from  C  as  a  center, 
with  a  radius  equal  to  3.34,  des 
cribe  the  circumference  cutting  the 
parabola  in  the  four  points,  M,  Mf,     C         „ 
M",  and  M'".     The  distance  of  M        ^ 
from  the  axis  of  X  is  +3.5,  of  M' 

it  is  +0.7,  of  M"  it  is  —1.5,  and  of  M'"  it  is  —2.7,  and 
these  are  the  four  roots  of  the  equation. 


H'! 


INTERSECTION  OF  LINES.  241 

Their  sum  is  0,  as  it  ought  to  be,  because  the  equation 
contains  no  third  power  of  y. 

2.  —  Find  the  roots  of  the  equation 

y*+y*+fy2+l2y—  72=0. 

This  equation  contains  the  third  power  of  y  ;  therefore 
this  geometrical  solution  will  not  apply  until  that  term  is 
removed. 

But  we  can  remove  that  term  by  putting 


(See  theory  of  transforming  equations  in  algebra). 
This  value  of  y  substituted  in  the  equation,  it  becomes 


and  this  equation  is  in  the  proper  form. 

Nowput  4—  4a=5f,—  86=9  J,  and  4(jft2—  a2—  62)=74jf  |. 

Whence      a=—  Jf,  6=—  |f,  and  -R=4.485. 

These  values  of  a  and  b  designate  the  point  Cf  for  the 
center  of  the  circle.  From  this  center,  with  a  radius 
=4.485,  we  strike  the  circumference,  cutting  the  parabola 
in  the  two  points  m  and  m'.  The  point  m  is  2J  units 
above  the  axis  A  X,  and  the  point  mf  is  —  2f  units  from 
the  same  line,  and  these  are  the  two  roots  of  the  equation. 
The  other  two  roots  are  imaginary,  shown  by  the  fact  that 
this  circumference  can  cut  the  parabola  in  two  points  only. 

If  we  conceive  the  circumference  of  a  circle  to  pass 
through  the  vertex  of  the  parabola  A,  then  will 

a?+b*=It2, 
and  this  supposition  reduces  the  general  equation  (F)  to 


Here  y=±0  will  satisfy  the  equation,  and  this  is  as  it 
should  be,  for  the  circumference  actually  touches  the  par 
abola  on  the  axis  of  X. 

Now  divide  this  last  equation  by  this  value  of  y,  and 
we  have 

21 


242  ANALYTICAL    GEOMETKY. 

Here  is  an  equation  of  the  third  degree,  referring  to  a 
parabola  and  a  circle  ;  the  circumference  cutting  the  par 
abola  at  its  vertex  for  one  point,  and  if  it  cuts  the  par 
abola  in  any  other  point,  that  other  point  will  designate 
another  root  in  equation  (G). 

It  is  possible  for  a  circle  to  touch  one  side  of  the  par 
abola  within,  and  cut  at  the  vertex  A  and  at  some  other 
point.  Therefore  it  is  possible  for  an  equation  in  the 
form  of  eq.  (G)  to  have  three  real  roots,  and  two  of  them 
equal. 

The  circumferences  of  most  circles,  however,  can  cut 
the  parabola  in  A  and  in  one  other  point,  showing  one 
real  root  and  two  imaginary  roots. 

Equation  (G)  can  be  used  to  effect  a  mechanical  solu 
tion  of  all  numerical  equations  of  the  third  degree,  in 
that  form.* 

"We  will  illustrate  this  by  one  or  two 

EXAMPLES. 

1. — Given  y3-f  4y=39,  to  find  the  value  of  j  by  construc 
tion.  (See  fig.  following  page) 

Put  4 — 4a=4,  and  86=39.     Whence  a=0,  and  &=4f. 

These  values  of  a  and  b  designate  the  point  C  on  the 
axis  of  Y  for  the  center  of  the  circle,  CA=4-J,  the  radius. 

The  circle  again  cuts  the  parabola  in  P,  and  PQ  mea 
sures  three  units,  the  only  real  root  of  the  equation. 

2. — Given  y3 — 75y=250,  to  find  the  values  of  y  by  con 
struction. 

When  the  co-efficients  are  large,  a  large  figure  is  re 
quired  ;  but  to  avoid  this  inconvenience,  we  reduce  the 
co-efficients,  as  shown  in  Chap.  2. 

*  Observe  that  the  second  term,  or  7/2,  in  a  regular  cubic  is  wanting. 
Hence,  if  any  example  contains  that  term,  it  must  be  removed  before  a 
geometrical  solution  can  be  given. 


INTERSECTION   OF  LINES. 


243 


Thus  put        y=nz. 
Then  the  equation  becomes 
nBz3—  75ftz= 


Now  take  n=5,  then  we  have 

z*—3z=2. 

In  this  last  equation  the  co-effi 
cients  are  sufficiently  small  to  apply  to  a  construction. 

Put  4—  4a=—  3,  and  86=2. 

Whence  a==^%,  and  6=J. 

These  values  of  a  and  b  designate  the  point  D  for  the 
center  of  the  circle.  D  A  is  the  radius. 

The  circle  cuts  the  parabola  in  t,  and  touches  it  in  T7, 
showing  that  one  root  of  the  equation  is  +2,  and  two 
others  each  equal  to  —  1. 

But  y=nz.     That  is,  #=5x2,  or—  5,  —5. 

Or  the  roots  of  the  original  equation  are  +10,  —  5,  —  5. 

When  an  equation  contains  the  second  power  of  the 
unknown  quantity,  it  must  be  removed  by  transforma 
tion  before  this  method  of  solution  can  be  applied. 

3.  —  Given  y3  —  48y=128  to  find  the  values  of  y  by  con 
struction.  Ans.     +8,  —  4,  —  4. 

4.  —  Given  y3  —  13y=  —  12,  to  find  the  values  of  y  by  con 
struction.  Ans.     +1,  +3,  and—  4. 

Conversely  we  can  describe  a  parobola,  and  take  any 
point,  as  H,  at  pleasure,  and  with  HA  as  a  radius,  de 
scribe  a  circle  and  find  the  equation  to  which  it  belongs. 

This  circle  cuts  the  parabola  in  the  points  m,  n  and  o, 
indicating  an  equation  whose  roots  are  +1,  -f  2.4,  and 
—3.4. 

We  may  also  find  the  particular  equation  from  the 
general  equation 


244  ANALYTICAL    GEOMETEY. 

observing  the  locality  of  H,  which  corresponds  to  a=3-3 
and  6=  —  1,  and  taking  these  values  of  a  and  b,  we  have 

f—  9.2?/=—  8, 
for  the  equation  sought. 

EEMAEKS     ON    THE   INTEEPEETATION  OF  EQUATIONS. 

In  every  science  it  is  important  to  take  an  occasional 
retrospective  view  of  first  principles,  and  the  conviction 
that  none  demand  this  more  imperatively  than  geometry 
will  excuse  us  for  reconsidering  the  following  truths  so 
often  in  substance,  if  not  in  words,  called  to  mind  before. 

An  equation,  geometrically  considered,  whatever  may  be  its 
degree,  is  but  the  equation  of  a  point,  and  can  only  designate  a 
point. 

Thus,  the  equation  y=ax-\-b  designates  a  point,  which 
point  is  found  by  measuring  any  assumed  value  which 
may  be  given  to  x  from  the  origin  of  co-ordinates  on  the 
axis  of  X,  and  from  that  extremity  measuring  a  distance 
represented  by  (ax-\-b)  on  a  line  parallel  to  the  axis  of  Y. 

The  extremity  of  the  last  measure  is  the  point  designated 
by  the  equation.  If  we  assume  another  value  for  x,  and 
measure  again  in  the  same  way,  we  shall  find  the  point 
which  now  corresponds  to  the  value  of  x.  Again,  as 
sume  another  value  for  x,  and  find  the  designated  point. 

Lastly,  if  we  connect  these  several  points,  we  shall  find 
them  all  in  the  same  right  line,  and  in  this  sense  the  equa 
tion  of  the  first  degree,  y=ax+b,  is  the  general  equation  of 
a  right  line,  but  the  right  line  is  found  by  finding  points 
in  the  line  and  connecting  them. 

In  like  manner  the  equation  of  the  second  degree 


only  designates  a  point  when  we  assume  any  value  for  x, 
(not  inconsistent  with  the  existence  of  the  equation),  and 
take  the  plus  sign.  It  will  also  designate  another  point 


INTERPRETATION   OF   EQUATIONS. 

when  we  take  the  minus  sign.  Taking  another  value  of 
x,  and  thus  finding  two  other  points,  we  shall  have  four 
points, — still  another  value  of  x  and  we  can  find  two  other 
points,  and  so  on,  we  might  find  any  number  of  points. 
Lastly,  on  comparing  these  points  we  shall  find  that  they 
are  all  in  the  circumference  of  the  same  circle,  and  hence  we 
say  that  the  preceding  equation  is  the  equation  of  a  circle. 
Yet  it  can  designate  only  one,  or  at  most,  two  points  at 
a  time. 

If  we  assume  different  values  for  y,  and  find  the  cor 
responding  values  of  x,  the  result  will  be  the  same  circle, 
because  the  x  and  y  mutually  depend  upon  each  other. 

Now  let  us  take  the  last  practical  example 

yz— 13y=— 12, 

and,  for  the  sake  of  perspicuity,  change  y  into  x,  then  we 
shall  have 

x3— 13x4-12=0. 

Now  we  can  suppose  ?/=0  to  be  another  equation ;  then 
will 

y=tf— 13x+12  (A) 

be  an  independent  equation  between  two  variables,  and 
of  the  third  degree. 

The  particular  hypothesis  that  y=0,  gives  three  values 
to  x,  (+1,  +3,  and — 4),  that  is,  three  points  are  designated: 
the  first  at  the  distance  of  one  unit  to  the  right  of  the 
axis  of  Y;  the  second  at  the  distance  of  three  units  on 
the  same  side  of  the  axis  of  Y;  and  the  third  point  four 
units  on  the  opposite  side  of  the  same  axis,  and  this  is  all 
the  equation  can  show  until  we  make  another  hypothesis. 

Again,  let  us  assume  ?/=5,  then  equation  (A)  becomes 

5=x3— 13x4-12,  or  x3— 13x4-7=0, 

and  this  is,  in  effect,  changing  the  origin  five  units  on  the 
axis  of  Y.  A  solution  of  this  last  equation  fixes  three 
other  points  on  a  line  parallel  to  the  axis  of  X. 

Again,  let  us  assume  J/=10,  then  equation  (A)  becomes 

x3— 13x4-2=0, 
21* 


246 


ANALYTICAL   GEOMETRY. 


7/=25. 

£=—1.1 

y=20. 

£=—0.40 

y=I5. 

£=—0.20 

</=10. 

£=+0.14 

and  a  solution  of  this  equation  gives  three  other  points. 
And  thus  we  may  proceed,  assigning  different  values 
to  y,  and  deducing  the  corresponding  values  of  £,  as  ap 
pears  in  the  following  table,  commencing  at  the  origin 
of  the  co-ordinates,  where  y=0,  and  varying  each  way. 
?/=30.0388  £=—2.2814  +4.1628  —2.0814 

+4.03 

+  3.80 

+  3.70 

+3.52 

£  I    r\  fr  er  __[__  o  o 

"When  ?/=0.  then  will  £=+1. 

#=—5  £=+1.66 

y=— 6.0388        £=+2.0814 

Taking  ?/=0,  a  solution  of  the 
equation  ?/=£3 — 13x+12,  gives  the 
three  points  a,  a,  a,  on  the  axis  of  X. 

Then  taking  y=5,  and  a  solution 
gives  three  points  6,  b,  b,  on  a  line  X 
parallel  to  the  axis  of  JT,  and  at  the 
distance  of  5  units  above  said  axis. 

Again,  taking  ?/=10,  and  another  solution  gives  the 
three  points  c,  c,  c.  Now  joining  the  three  points  (a,  b,  c,) 
(a,  b,  c),  and  (a,  b,  c),  we  shall  have  apparently  three  curves 
corresponding  to  the  equation  of  the  third  degree,  and 
thus,  we  might  hastily  conclude  that  every  equation  of 
the  third  degree  would  give  three  curves,  and  every  equa 
tion  of  the  fourth  degree  four  curves,  etc.,  etc.,  but  this  is 
not  true. 

If  we  continue  finding  points  as  before,  we  shall  find 
that  the  three  curves  (a,  b,  c,)  (a,  b,  c,)  and  (a,  b,  c,)  are 
but  different  portions  of  the  same  curve,  and  wre  can  now 
venture  to  draw  this  general  conclusion  : 

That  in  an  equation  involving  y,  the  ordinate,  to  the  first  power, 


INTERPRETATION  OF  EQUATIONS.     247 

and  the  abscissa,  x,  to  the  third  power,  the  axis  of  X,  or  lines 
parallel  to  that  axis,  may  cut  the  curve  in  three  points. 

From  analogy,  we  also  infer  that  if  we  have  an  equa 
tion  involving  x  to  the  fourth  power,  the  axis  of  X,  or 
its  parallels,  Will  cut  the  curve  in  four  points  ;  and  if  we 
have  an  equation  involving  x  to  the  fifth  power,  that  axis  or 
its  parallels  will  cut  the  curve  in  five  points,  and  so  on. 

In  the  equation  under  consideration,  (y=x* — 13x+12), 
if  we  assume  y  greater  than  30.0388,  or  less  than  — 6.0388, 
we  shall  find  that  two  values  of  x  in  each  case  will  be 
come  imaginary,  and  on  each  side  of  these  limits  the 
parallels  to  JTwill  cut  the  curve  only  in  one  point. 

Two  points  vanish  at  a  time,  and  this  corresponds  with 
the  truth  demonstrated  in  algebra,  "  that  imaginary  roots 
enter  equations  in  pairs." 

The  points  m,  m,  the  turning  points  in  the  curve,  are 
called  maximum  points,  and  can  be  found  only  by  approx 
imation,  using  the  ordinary  processes  of  computation, 
but  the  peculiar  operation  of  the  calculus  gives  these 
points  at  once. 

To  find  the  points  in  the  curve  we  might  have  assumed 
different  values  of  x  in  succession,  and  deduced  the  cor 
responding  values  of  y,  but  this  would  have  given  but 
one  point  for  each  assumption ;  and  to  define  the  curve 
with  sufficient  accuracy,  many  assumptions  must  be  made 
with  very  small  variations  to  x.  ,  We  solved  the  equa 
tions  approximately  and  with  great  rapidity  by  means  of 
the  circle  and  parabola  as  previously  shown. 

We  conclude  this  subject  by  the  following  example : 

Let  the  equation  of  a  curve  be 

(a2— x2)(x— 6)2=ry, 

from  which  we  are  required  to  give  a  geometrical  deline 
ation  of  the  curve.     From  the  equation  we  have 


248  ANALYTICAL    GEOMETRY. 

The  following  figure  represents  the  curve  which  will 
be  recognized  as  corresponding  to  the  equation,  after  a 
little  explanation. 

If  £=0,  then  y  becomes  infinite, 
and  therefore  the  ordinate  at  A  is  an 
asymptote  to  the  curve.  If  AB=b, 
and  P  be  taken  between  A  and  _B, 
then  FM  and  Pm  will  be  equal,  and 
lie  on  different  sides  of  the  abscissa 
AP.  If  x=b,  then  the  two  values  of 
y  vanish,  because  x — 6=0;  and  consequently,  the  curve 
passes  through  B,  and  has  there  a  duplex  point.  If  AP 
be  taken  greater  than  AB,  then  there  will  be  two  values 
of  y,  as  before,  having  contrary  signs,  that  value  which 
was  positive  before,  now  becomes  negative,  and  the  nega 
tive  value  becomes  positive.  But  if  AD  be  taken  =a, 
and  P  come  to  D,  then  the  two  values  of  y  vanish,  because 
+/az  Xv—Qf  And  if  A  P  is  taken  greater  than  AD,  then 
a2 — x2  becomes  negative,  and  the  value  of  y  impossible  ; 
and  therefore,  the  curve  does  not  extend  beyond  D. 

If  x  now  be  supposed  negative,  we  shall  find 

y=  zb^a2"-^?  x  (b  +x)~-x? 

If  x  vanish,  both  these  values  of  y  become  infinite,  and 
consequently,  the  curve  has  two  infinite  arcs  on  each  side 
of  the  asymptote  AK.  If  x  increase,  it  is  plain  y  dimin 
ishes,  and  if  x  becomes  = — a,y  vanishes,  and  consequently 
the  curve  passes  through  E,  if  AE  be  taken  =  AD,  on 
the  opposite  side.  If  x  be  supposed,  numerically,  greater 
than  — a,  then  y  becomes  impossible  ;  and  no  part  of  the 
curve  can  be  found  beyond  E.  This  curve  is  the  conchoid 
of  the  ancients. 


STRAIGHT   LINES   IN   SPACE.  249 

CHAPTER  VUL 
STRAIGHT  LIKES  DT  SPACE. 

Straight  lines  in  one  and  the  same  plane  are  referred 
to  two  co-ordinate  axes  in  that  plane,  — but  straight  lines 
in  space  require  three  co-ordinate  axes,  made  by  the  inter 
section  of  three  planes. 

To  take  the  most  simple  view  of  the  subject,  conceive 
a  horizontal  plane  cut  by  a  meridian  plane,  and  by  a  per 
pendicular  east  and  west  plane. 

The  common  point  of  intersection  we  shall  call  the 
origin  or  zero  point,  and  we  might  conceive  this  point  to 
be  the  center  of  a  sphere,  and  about  it  will  be  eight  quad 
rangular  spaces  corresponding  to  the  eight  quadrants  of 
a  sphere,  which  extended,  would  comprise  all  space. 

The  horizontal  east  and  west  line  of  intersection  of 
these  planes,  we  shall  call  the  axis  of  X.  The  horizon 
tal  intersection  in  the  direction  of  the .  meridian,  the  axis 
of  T;  and  that  perpendicular  to  it  in  the  plane  of  the 
meridian,  the  axis  of  Z.  Distances  estimated  from  the 
zero  point  horizontally  to  the  right,  as  we  look  towards 
the  north,  we  shall  designate  as  plus,  to  the  left  minus. 

Distances  measured  on  the  axis  of  Y  and  parallel 
thereto,  towards  us  from  the  zero  point,  we  shall  call  plus; 
those  in  the  opposite  direction  will  therefore  be  minus. 
Perpendicular  distances  from  the  horizontal  plane  up 
wards  are  taken  as  plus,  downward  minus. 

The  horizontal  plane  is  called  the  plane  of  xy,  the  me 
ridian  plane  is  designated  as  the  plane  of  yz,  and  the  per 
pendicular  east  and  west  plane  the  plane  of  xz. 

Now  let  it  be  observed  that  x  will  be  plus  or  minus,  ac 
cording  to  its  direction  from  the  plane  of  yz,  y  will  be 
plus  or  minus,  according  to  its  direction  from  the  plane 


250  ANALYTICAL    GEOMETRY. 

xZy  and  z  will  be  plus  or  minus,  according  as  it  is  above  or 
below  the  horizontal  place  xy. 


PROPOSITION   I. 

To  jmd  the  equation  of  a  straight  line  in  space. 

Conceive  a  straight  line  passing  in  any  direction  through 
space,  and  conceive  a  plane  coinciding  with  it,  and  per 
pendicular  to  the  plane  xz.  The  intersection  of  this 
plane  with  the  plane  xz,  will  form  a  line  on  the  plane  xz, 
and  this  is  said  to  be  the  projection  of  the  line  on  the 
plane  xz,  and  the  equation  of  this  projected  line  will  be 
in  the  form 

x=az-\-7r.     (Chap.  1,  Prop.  1.) 

Conceive  another  plane  coinciding  with  the  proposed 
line,  and  perpendicular  to  the  plane  yz,  its  intersection 
with  the  plane  yz  is  said  to  be  the  projection  of  the  line 
on  the  plane  yx,  and  the  equation  of  this  projected  line 
is  in  the  form 


These  two  equations  taken  together  are  said  to  be 
equations  of  the  line,  because  the  first  equation  is  a  gen 
eral  equation  for  all  lines  that  can  be  drawn  in  the  first 
projecting  plane,  and  the  second  equation  is  a  general 
equation  for  all  lines  that  can  be  drawn  in  the  second 
projecting  plane  ;  therefore  taken  together,  they  ex 
press  the  intersection  of  the  two  planes,  which  is  the  line 
itself. 

For  illustration,  we  give  the  following  example  :  Construct 
the  line  whose  equations  are 


=32—  2 


STRAIGHT   LINES   IN   SPACE.  251 

Make  2=0,  then  rc=l,  and  ?/= — 2. 
Now  take  J.P=1,  and  draw  Pm 
parallel  to  the  axis  of  Y,  making 
Pm= — 2 ;  then  m  is  the  point  in 
the  plane  xy,  through  which  the  /  T7~X 
line  must  pass.  /- IL. 

Now  take  z  equal  to  any  num 
ber  at  pleasure,  say  1,  then  we  shall 
have  z=3  and  y=l. 

Take  J.P^=3,  P'm'=-f  1,  and  from  the  point  mr  in  the 
plane  xy  erect  mrn  perpendicular  to  the  plane  xy>  and 
make  it  equal  to  1,  because  .we  took  2=1,  then  n  is  an 
other  point  in  the  line.  Draw  n  m  and  produce  it,  and  it 
will  be  the  line  designated  by  the  equations. 

PROPOSITION   II. 

To  find  the  equation  of  a  straight  line  lohich  shall  pass 
through  a  given  point. 

Let  the  co-ordinates  of  the  given  point  be  represented 
by  #',  j/ ,  zf. 

The  equations  sought  must  satisfy  the  general  equa 
tions 

x=az+x. 


The  equations  corresponding  to  the  given  point  are 

x'=az'+K.  y'=bzf+fi. 

Subtracting  eq.  (1)  from  these,  respectively,  we  have 

xr — x=a(zf — 2),  and  yf — y=b(zf — z\ 
the  equations  required. 

PROPOSITION    III. 

To  find  the  equations  of  a  straight  line  which  shall  pass 
through  two  given  points. 


252  ANALYTICAL    GEOMETRY. 

Let  the  co-ordinates  of  the  second  point  be  x",  y",  z", 
Now  by  the  second  proposition,  the  equations  which  ex 
press  the  condition  that  the  line  passes  through  the  two 

points,  will  be 

x»—X'=a(z"—  z')9 

Alid  y»—y>  =  b(z"—Zf}. 

"Whence  a-^JZ^,  b=y"~~y'-. 

2"_2/  '          Z»—Z> 

Substituting  the  values  of  a  and  6  in  the  equations  of  a 
line  passing  through  a  single  point  (Prop.  2,)  we  have 


for  the  equations  required. 


PROPOSITION   IV. 

To  find  the  condition  under  ivhich  two  straight  lines  intersect 
in  space,  and  the  co-ordinates  of  the  point  of  intersection. 
Let  the  equation  of  the  lines  be 


x=a'z+n'.  y=bfz+pf. 

If  the  two  lines  intersect,  the  co-ordinates  of  the  com- 
mojo.  point,  which  may  be  denoted  by  x9  y,  z,  will  satisfy 
all  of  these  four  equations,  therefore  by  subtraction,  we 
have 


Whence,  by  eliminating  2,  we  find 

7T--  7r'=/9—  p' 


which  is  the  condition  under  which  two  lines  intersect. 

Now  2=7r      ^  and  this  value  of  z  being   substituted 

' 


a  —  a 


in  the  first  equations,  we  obtain 


an'+~ctn  '      ,          b8'—b> 
and    y—  ^ 


a — a' 


STRAIGHT   LINES  IN   SPACE. 


253 


for  the  value  of  the  co-ordinates  of  the  point  of  inter 
section. 

Cor. — If  a~a',  the  denominators  in  the  second  mem 
ber  will  become  0,  making  x  and  y  infinite ;  that  is,  the 
point  of  intersection  is  at  an  infinite  distance  from  the 
origin,  and  the  lines  are  therefore  parallel. 

PROPOSITION    V.— PROBLEM. 

To  express  analytically  the  distance  of  a  given  point  from 
the  origin. 

Let  P  be  the  given  point  in 
space  ;  it  is  in  the  perpendicular 
at  the  point  N9  which  is  in  the 
plane  xy. 

The  angle  AMN=9Q°.  Also, 
the  angle  JJVP=90°. 

Let  AM—Xj  MN=y,  NP=z. 

Then  JJ?2=, 
But  T~P2— 


"Now  if  we  designate  AP  by  r,  we  shall  have 

r2=x2-f;?/2+22 
for  the  expression  required. 

PROPOSITION  YI.— PROBLEM 

To  express  analytically  the  length  of  a  line  in  space. 

Let  PP'=D  be  the  line  in  question.     z 

Let  the  co-ordinates  of  the  point  P 
be  x,  ?/,  z9  and  of  the  point  Pf  be  x'9 
y',  z'. 

Now    MM'=x'—x=NQ. 
QN'=y'—y. 


22 


254  ANALYTICAL    GEOMETRY. 

In  the  triangle  PEPf  we  have 


Or  &=(xf—x)*+(yf—  y^+y—  zf,          (1) 

which  is  the  expression  required. 

SCHOLIUM.  —  If  through  one  extremity  of  the  line,  as  P,  we 
draw  PA  to  the  origin,  and  from  the  other  extremity  P',  we  draw 
P'$  parallel  and  equal  to  PA,  and  draw  AS,  it  will  be  parallel  to 
PPf,  and  equal  to  it,  and  this  virtually  reduces  this  proposition  to 
the  previous  one.  This  also  may  be  drawn  from  the  equation,  for 
if  A  is  one  extremity  of  the  line,  its  co-ordinates  x}  yt  and  z  are 
each  equal  to  zero,  and 


PROPOSITION    VII.—  PROBLEM. 

To  find  the  inclination  of  any  line  in  space  to  the  three  axes. 
From  the  origin  draw  a  line         z 
**  parallel  to  the  given  line  ;  then 
the  inclination  of  this  line  to  the 
axes  will  be  the  same  as  that  of 
the  given  line. 

The  equations  for  the  line  pass- 
ing  from  the  origin  are 

x=az,    and    y=bz.  (1) 

Let  X  represent  the  inclination  of  this  line  with  the 
axis  of  x,  Y  its  inclination  with  the  axis  of  y,  and  Z  its 
inclination  with  the  axis  of  z. 

The  three  points  P,  N7  M,  are  in  a  plane  which  is  par 
allel  to  the  plane  zy,  and  A  M  is  a  perpendicular  between" 
the  two  planes.  AMP  is  a  right-angled  triangle,  the 
right  angle  being  at  M. 

Let  AP—r  and  A  M=x.  Then,  by  trigonometry,  we 
have 

As  r  :  sin.  90°  :  :  x  :  cos.  X.     Whence  x=r  cos.  X. 

Also,  as  r  :  sin.  90°  :  :  y  :  cos.  Y.     Whence  y=r  cos.  F. 


STRAIGHT   LINES  IN   SPACE.  255 

Also,  as  r  :  sin.  90°  :  :  z  :  cos.  Z.     Whence  z—r  cos.  Z. 
From  Prop.  5  we  have 

r*  =  x2+y2+Z*.  (2) 

Substituting  the  values  of  x,  y,  and  z,  as  above,  we 
have 

r2=r2  cos.2  JT"-f  r2  cos.2  Y+r2  coa.2Z. 
Dividing  by  r2  will  give 

cos.sJT-fcos.2  r+cos.2£=l,          (3) 

an  equation  which  is  easily  called  to  mind,  and  one  that 
is  useful  in  the  higher  mathematics. 

If  in  eq.  (2)  we  substitute  the  values  of  x2  and  y2  taken 
from  eq.  (1),  we  shall  have 


But  we  have  three  other  values  of  r2  as  follows  : 

r2  =     x*—  ,     r2  =     ^2_       and  r2=  _  ?__. 
cos.2  X  cos.2  Y  cos.2  Z 

"Whence  ~^^ 

cos.  A 


And  _=±v'1+a8+52.  (7) 

COS.  ^ 

In  eq.  (5)  put  the  value  of  x  drawn  from  eq.  (1),  and  in 
eq.  (6)  the  value  of  y  from  eq.  (1),  and  reduce,  and  we 
shall  obtain 


The  analytical  expressions 
for  the  inclination  of  a  line 
in  space  to  the  three  co-or 
dinates. 


COS.  ^= 


The  double  sign  shows  two  angles  supplemental  to 
each  other,  the  plus  sign  corresponds  to  the  acute  angle, 
and  the  minus  sign  to  the  obtuse  angle. 


256  ANALYTICAL    GEOMETRY. 

PKOPOSITION    VIII. 

To  find  the  inclination  of  two  lines  in  terms  of  their  sepa 
rate  inclinations  to  the  axes. 

Through  the  origin  draw  two  lines  respectively  paral 
lel  to  the  given  lines.  An  expression  for  the  cosine  of 
the  angle  between  these  two  lines  is  the  quantity  sought. 

Let  AP  be  parallel  to  one  of  the  given  lines,  and  AQ 
parallel  to  the  other.  The  angle  PA  Q  is  the  angle  sought. 

Let  the  equations  of  one  of  these  lines  be 

x=az,  y=bz, 

and  of  the  other 

x*=a'z'9  y'=b'z'. 

Let  AP=r,  AQ=rf,  PQ=D,  and  the  angle  PAQ=  V. 

Now  in  plane  trigonometry  (Prop.  8,  p.  260,  Geom.,) 
we  have 


2rr' 
From  Prop.  6  we  have 


Expanding  this,  it  becomes 


—2x'x—2y'y—2z'z. 
But  by  Prop.  5  we  have 


Q 

"  P 


and  xf*+yf2+z'2=r/2. 

Whence      2x'x+2y'y+2z'z=r*+r's— D*. 
This  equation  applied  to  eq.  (I)  reduces  it  to 

cos.  V 


rr 


But  r  and  rr  may  have  any  values  taken  at  pleasure; 
their  lengths  will  have  no  effect  on  the  angle  V.  There 
fore,  for  convenience,  we  take  each  of  them  equal  to 
unity. 

Whence  cos.  V=*xrx+y'y+zfz.  (2) 


STKAIGHT   LINES   IN   SPACE.  257 


But  in  Prop.  7  we  found  that  x=rGos.^  y=r  cos.  Y", 
etc.,  and  that  xr—r'  cos.^T',  yf=rj  cos.  Y',  etc.  ;  and  since 
we  have  taken  r=l  and  r'=l,  x=cos.  X,  etc.,  and  x'  = 
cos.JT',  etc.  Hence 

cos.  7=cos.Xcos.  J^'+cos.  Fcos.  Y'+cos.^cos.Z'.  (3) 

But  by  Prop.  7  we  have 

CL 

cos.  JT=  —  .  r=.    and  cos.  X= 


Substituting  these  values  in  eq.  (3)  we  have 

l+ao'+W 

cos.  y=- 


for  the  expression  required. 

The  cos.  V  will  be  plus  or  minus,  according  as  we  take 
the  signs  of  the  radicals  in  the  denominator  alike  or  un 
like.  The  plus  sign  corresponds  to  an  acute  angle,  the 
minus  sign  to  its  supplement. 

Cor.  1.—  If  we  make  7=90°,  then  cos.  7=0,  and  the 
equation  becomes 


which  is  the  equation  of  condition  to  make  two  lines  at 
right  angles  in  space. 

Cor.  2.  —  -If  we  make  7=0,  the  two  straight  lines  will 
become  parallel,  and  the  equation  will  become 

±1= 


Squaring,  clearing  of  fractions,  and  reducing,  we  shall 
find 

(a!  _a)«+  (fi/_  &)2+  (aV  —a'b?=  0. 

Each  term  being  a  square,  will  be  positive,  and  there 
fore  the  equation  can  only  be  satisfied  by  making  each 
term  separately  equal  to  0. 

"Whence  a'=a,  b'  =6,  and  abf=afb. 

The  third  condition  is  in  consequence  of  the  first  two. 
22*  E 


258  ANALYTICAL    GEOMETRY. 

CHAPTER  IX. 
ON  THE  EQUATION  OF  A  PLANE. 

An  equation  which  can  represent  any  point  in  a  line 
is  said  to  be  the  equation  of  the  line. 

Similarly,  an  equation  which  can  represent  or  indicate 
any  point  in  a  plane,  is,  in  the  language  of  analytical  ge 
ometry,  the  equation  of  the  plane. 

PROPOSITION  I. 

To  find  the  equation  of  a  plane. 

Let  us  suppose  that  we  have  a  plane  which  cuts  the 
axes  of  JT,  Y  and  Z  at  the  points  JB,  C  and  D,  respec 
tively  ;  then,  if  these  points  be  connected  by  the  straight 
lines  BCj  CD  and  DJ3,  it  is  evi 
dent  that  these  lines  are  the  inter 
sections  of  the  plane  with  the 
planes  of  the  co-ordinate  axes. 

Now  a  plane  may  be  conceived 
as  a  surface  generated  by  moving  a 
straight  line  in  such  a  manner  that 
in  all  its  positions  it  shall  be  parallel  to  its  first  position 
and  intersect  another  fixed  straight  line.  Thus  the  line 
DC,  so  moving  that  in  the  several  positions,  D' 'C",  D"C", 
etc.,  it  remains  parallel  to  DO  and  constantly  intersects 
DjB,  will  generate  the  plane  determined  by  the  points  D, 
C  and  B. 

The  line  DB  being  in  the  plane  xy,  its  equations  are 

2/=0,  z=mx+b,  (1) 

and  for  the  line  DC  we  have 

z=0,  z=ny+b.  (2) 

The  plane  passed  through  the  line  U  C'  parallel  to  the 


EQUATION    OF   A   PLANE.  259 

plane  zy,  cuts  the  axis  of  JTat  the  point  p.     Denoting  Ap 
by  Cj  the  equations  of  the  line  DfCf  become 

x=c,  z=ny+b'.  (3) 

It  is  obvious  that  eqs.  (3)  can  be  made  to  represent  the 
moving  line  in  all  its  positions  by  giving  suitable  values 
to  c  and  bf,  and  that,  for  any  one  of  its  positions,  the  co 
ordinates  of  its  intersection  with  the  line  DB  must  satisfy 
both  eqs.  (1)  and  (3).  That  is,  c  and  b',  in  the  first  and 
second  of  eqs.  (3),  must  be  the  same  as  x  and  £,  respec 
tively,  in  the  second  of  eqs.  (1).  Hence 
b'=z — ny,  and  b'=mx+b. 

Equating  these  two  values  of  b',  we  have 

z — ny=mx+b, 
or  z=mx+ny+b.  (4) 

This  equation  expresses  the  relation  between  the  co-or 
dinates  x,  y  and  z  for  any  point  whatever  in  the  plane 
generated  by  the  motion  of  the  line  DC,  and  is,  there 
fore  the  equation  of  this  plane. 

Cor.  1. — Every  equation  of  the  first  degree  between 
three  variables,  by  transposition  and  division,  may  be  re 
duced  to  the  form  of  eq.  (4),  and  will,  therefore,  be  the 
equation  of  a  plane. 

Cor.  2. — In  eq.  (4),  m  is  the  tangent  of  the  angle  which 
the  intersection  of  the  plane  wTith  the  plane  xz  makes  with 
the  axis  of  X,  n  the  tangent  of  the  angle  that  the  inter 
section  with  the  plane  yz  makes  with  the  axis  of  Y,  and 
b  the  distance  from  the  origin  to  the  point  in  which  the 
plane  cuts  the  axis  of  Z. 

Hence,  if  any  equation  of  the  first  degree  between  three  vari 
ables  be  solved  with  respect  to  one  of  the  variables,  the  co-effi 
cient  of  either  of  the  other  variables  denotes  the  tangent  of  the 
angle  that  the  intersection  of  the  plane  represented  by  the  equa- 
tionjjoith  the  plane  of  the  axes  of  the  first  and  second  variables, 
makes  with  the  axis  of  the  second  variable. 


260 


ANALYTICAL  GEOMETKY. 


SCHOLIUM.  —  If  we  assume 


ro=—       »=-:.     =- 


C  C 

and  substitute  these  values  in  eq.  (4),  it  will  become,  by  reduction 
and  transposition, 


which  is  the  form  under  which  the  equation  of  the  plane  is  very 
often  presented. 

From  this  equation  we  deduce  the  following  general  truths : 

First. — If  we  suppose  a  plane  to  pass  through  the  origin  of  the 
co-ordinates  for  this  point,  #=0,  y=0,  and  2= 0,  and  these  values 
substituted  in  the  equation  of  the  piano  will  give  D=Q  also.  There 
fore,  when  a  plane  passes  through  the  origin  of  co-ordinates,  the 
general  equation  for  the  plane  reduces  to 
Ax+£y+Cz=Q. 

/Second. — To  find  the  points  in  which  the  plane  cuts  the  axes,  we 
reason  thus : 

The  equation  of  the  plane  must  respond 
to  each  and  every  point  in  the  plane ;  the 
point  P,  therefore,  in  which  the  plane  cuts 
the  axis  of  X,  must  correspond  to  y=0 
and  2=0,  and  these  values,  substituted  in 
the  equation,  reduces  it  to 


Or  *=—£!: 

A 

For  the  point  Q  we  must  take  x=Q  and  «=0. 
And  y= — ~L.~OQf. 

For  the  point  R,  z= — _=OJ2. 

Third. — If  we  suppose  the  plane  to  be  perpendicular  to  the  plane 
XY,  PR',  its  intersection  with,  or  trace  on,  the  plane  XZ,  must  be 
drawn  parallel  to  OZ,  and  the  plane  will  meet  the  axis  of  Z  at  the 
distance  infinity.  That  is,  OR,  or  its  equal,  (— -  ) ,  must  be  infi 
nite,  which  requires  that  (7=0,  which  reduces  the  general  equation 
of  the  plane  to 


EQUATION   OF  A  PLANE.  261 

Ax-\-  By  -\-D-Qj 

which  is  the  equation  of  the  trace  or  line  PQ  on  the  plane  XY. 
If  the  plane  were  perpendicular  to  the  plane  ZX,  the  line  0  Q,  or 
its  equal,  (  --  j,  must  be  infinite,  which  requires  that  .#=0,  and 

this  reduces  the  general  equation  to 


which  is  the  equation  for  the  trace  PR,  and  hence  we  may  conclude 
in  general  terms, 

That  when  a  plane  is  perpendicular  to  any  one  of  the  co-ordinate 
planes,  its  equation  is  that  of  its  trace  on  the  same  plane. 

PKOPOSTION  II.—  PEOBLEM. 

To  find  the  length  of  a  perpendicular  drawn  from  the  origin 
to  a  plane,  and  to  find  its  inclination  with  the  three  co-ordinate 
axes. 

Let  JRPQ  be  the  plane,  and  from  the 
origin,  0,  draw  Op  perpendicular  to  the 
plane  ;  this  line  will  be  at  right-angles 
to  every  line  drawn  in  the*  plane  from 
the  pointy. 

Whence  Op  §=90°,  Op^=90°,  and 
QpP=90°. 

Let  Op—  p. 

Designate  the  angle  pOP  by  X,  pOQ  by  Y9  and 
by^. 

By  the  preceding  scholium  we  learn  that 

=-,  06=-      and  OR=- 


A,  13,  C  and  D  being  the  constants  in  the  equation  of  a 
plane. 

Now,  in  the  right-angled  triangle  OpP9  we  have 
OP  :  1  :  :  Op  :  cos.  X. 

That  is,  —-.  :l::p:  cos.  Z.  (1) 

A 


262  ANALYTICAL    GEOMETRY. 


The  right-angled  triangle  OpQ  gives 

— :?:!::«:  cos.  Y. 
B 

The  right-angled  triangle  OpR  gives 
_2>. 
C" 
Proportion  (1)  gives  us 


— P.  :l::pi  cos.  r.  (2) 

_B 


•— :  1:  :2>:  coa.Z.  0) 

0 


5s' 


(6) 


(2)  gives  cos.2  Y=^B' 

and  (3)  gives  cos.2  ^==^  (7: 

Adding  these  three  equations,  and  observing  that  the 
sum  of  the  first  members  is  unity,  (Prop.  7,  Chap.  8),  and 
we  have 


(5) 


Whence  y?=±  —    P  CO 


This  value  of  p  placed  in  eqs.  (4),  (5)  and  (6),  by  re 
duction,  will  give 

cos.  JT=  =b  A     __  (8) 


COS.  F=dr  __  .  (9) 


cos.  Z=± —  — .  (10) 

Expressions  (7),  (8),  (9)  and  (10)  are  those  sought. 

PROPOSITION   III.— PROBLEM. 

To  find  the  analytical  expressions  for  the  inclination  of  a 
plane  to  the  three  co-ordinate  planes  respectively. 


EQUATION    OF    A   PLANE. 


263 


Let  Ax+Ey+  Cz+D=0  be  the  equa 
tion  of  the  plane,  and  let  PQ  represent 
its  line  of  intersection  with  the  co-ordi 
nate  plane  (xy}. 

From  the  origin,  0,  draw  OS  per 
pendicular  to  the  trace  PQ.  Draw  pS. 
OpS  is  a  right-angled  triangle,  right- 
angled  at  p,  and  the  angle  OSp  measures  the  inclination 
of  the  plane  with  the  horizontal  plane  (xy].  Our  object 
is  to  find  the  angle  OSp. 

In  the  right-angled  triangle  POQ  we  have  found 

D    '***••     JD          * 


OP=- 


D 


Whence 


Now  PS,  a  segment  of  the  hypothenuse  made  by  the 
perpendicular  OS,  is  a  third  proportional  to  PQ  and  PO. 
Therefore 


:  —B  :  :  —  -r  :  PS= 
A 


Or 


The  other  segment,  QS,  is  a  third  proportional  to  PQ 
and  OQ.     Therefore 

D     . D  D     ™ 


. 

AB 


Or 


AD 


But  the  perpendicular,  OS,  is  a  mean  proportional  be 
tween  these  two  segments.     Therefore  we  have 

OS= — 


!N"ow,  by  simple  permutation,  we  may  conclude  that  the 
perpendicular  from  the  origin  0  to  the  trace  PR  is 


264  ANALYTICAL   GEOMETRY. 

D 


and  that  to  the  trace  QR  is 

D 


"We  shall  designate  the  angle  which  the  plane  makes 
with  the  plane  of  (xy]  by  (xy)<  and  the  angle  it  makes 
with  (xz)  by  (xz\  and  that  with  (yz}  by  (yz). 

!N"ow  the  triangle  OpS  gives 

OS :  sin.  90°  :  :  Op  :  sin.  OSp. 

D  D 

That  is,     ,         —  :  1  :  : 
' 


Whence 


But  by  trigonometry  we  know  that  cos.2=l — sin.2. 

A  24-  7?2  r72 

mence  003.^)=!---=,  eta 

Whence      c 


C2 

_u  A 

cos.(yz) 


C2 

>  Expressions  sought. 


C2  ^ 
Squaring,  and  adding  the  last  three  equations,  we  find 

cos.2(r?/) + cos.2(#2)  -f  cos.2(  yz)=l. 

That  is,  the  sum  of  the  squares  of  the  cosines  of  the  three 
angles  which  a  plane  forms  with  the  three  co-ordinate  planes, 
is  equal  to  radius  square,  or  unity. 


EQUATION    OF   A    PLANE.  265 

PKOPOSITION    IV.—  PROBLEM. 

To  find  the  equation  of  the  intersection  of  two  planes. 

Let  Ax+By+Cz+D=Q,  (1) 

A'x+B'y+  C'2+jD'=0,  (2) 

be  the  equations  of  the  two  planes. 

If  the  two  planes  intersect,  the  values  of  £,  y  and  z 
will  be  the  same  for  any  point  in  the  line  of  intersection. 
Hence,  we  may  combine  the  equations  for  that  line. 

Multiply  eq.  (1)  by  Cf  and  eq.  (2)  by  (7,  and  subtract 
the  products,  and  we  shall  have 

(A  C'—  A1  C}x+  (B  O—  Bf  C)y+  (D  C'  —  D'  <?)=  0, 
for  the  equation  of  the  line  of  intersection  on  the  plane 
(xy).  If  we  eliminate  y  in  a  similar  manner,  we  shall 
have  the  equation  of  the  line  of  intersection  on  the  plane 
(xz)  ;  and  eliminating  x  will  give  us  the  equation  of  the 
line  of  intersection  on  the  plane  (yz). 

PKOPOSITION    Y.—  PROBLEM. 

To  find  the  equation  to  a  perpendicular  let  fall  from  a  given 
point  (x',  y',  z'  ,)  upon  a  given  plane. 

As  the  perpendicular  is  to  pass  through  a  given  point, 
its  equations  must  be  of  the  form 

x—x'=a(z—z')9  (1) 

y—yf=b(z—z'\  (2> 

in  which  a  and  b  are  to  be  determined. 
The  equation  of  the  plane  is 


The  line  and  the  plane  being  perpendicular  to  each  other, 
by  hypothesis,  the  projection  of  the  line  and  the  trace  of 
the  plane  on  any  one  of  the  co-ordinate  planes  will  be 
perpendicular  to  each  other. 

For  the  traces  of  the  given  plane  on  the  planes  (xz)  and 
(yz\  we  have  Ax+  Cz+D=Q  and  By+  Cz+D=0. 
23 


266  ANALYTICAL    GEOMETKY. 

From  the  former      x— — —z — _.  (3) 

A      A 

From  the  latter        y=— ~z— :?.  (±) 

JD        JD 

ISTow  eqs.  (1)  and  (3)  represent  lines  which  are  at  right 
angles  with  each  other. 

Also,  eqs.  (2)  and  (4)  represent  lines  at  right  angles 
with  each  other. 

But  when  two  lines  are  at  right  angles,  (Prop.  5,  Chap. 
1),  and  a  and  a1  are  their  trigonometrical  tangents,  we 
must  have  (aa'-f-l=0). 

That  is,  — <£+l=0,  or  a=^. 

^JL  O 

T> 

Like  reasoning  gives  us  b= ,  and  these  values  put  in 

C 

eqs.  (1)  and  (2)  give 

x—x'=~(z—z')    |for 

,;       y-y>=*(z-z.) 


PROPOSITION    VI.— PROBLEM. 

To  find  the  angle  included  by  two  planes  given  by  their 
equations. 

Let  Ax+By+Cz+D=09  (1) 

And  A'x+3y'+C'z+iy=09  (2) 

be  the  equations  of  the  planes. 

Conceive  lines  drawn  from  the  origin  perpendicular  to 
each  of  the  planes.  Then  it  is  obvious  that  the  angle 
contained  between  these  two  lines  is  the  supplement  of  the 
inclination  of  the  planes.  But  an  angle  and  its  supple 
ment  have  numerically  the  same  trigonometrical  ex 
pression. 


EQUATION    OF    A   PLANE.  267 

Designate  the  angle  between  the  two  planes  by  V,  then 
Proposition  8,  in  the  last  chapter  gives 
Jr  I+aa'+bb' 

COS.    V=  —  -  —  —  -  .      (3) 


The  equations  of  the  two  perpendicular  lines  from  the 
origin  must  be  in  the  form 

x=az,  y=bz, 

x—a'z  y=b'z. 

But  because  the  first  line  is  perpendicular  to  the  first 
plane,  we  must  have 

a=4,      and      6=:?,     (Prop.  5.) 
o  u 

And  to  make  the  second  line  perpendicular  to  the  sec 
ond  plane  requires  that 

«'-£     and     V-». 

^These  values  of  a,  6,  and  a',  V,  substituted  in  eq.  (3) 
will  give,  by  reduction, 

cos.  T^+ 


for  the  equation  required. 

Cor.  —  When  two  planes  are  at  right  angles,  cos.  F=0, 
which  will  make 

AA'+BB'+ 


PROPOSITION   VII.—  PROBLEM. 

To  find  the  inclination  of  a  line  to  a  plane. 
Let  MN  be  the  plane  given  by  its  equation 


and  let  PQ  be  the  line  given  by  its  equations 


268  ANALYTICAL    GEOMETRY. 

x=az+a. 


1VP 


Take  any  point  P  in  the  given  line, 
and  let  fall  PR,  the  perpendicular,  up 
on  the  plane  ;  RQ  is  its  projection  on 
the  plane,  and  PQR,  which  we  will 
denote  by  F,  is  obviously  the  least  an- 
gle  included  between  the  line  and  the  plane,  and  it  is  the 
angle  sought. 

Let  x—afz+7r'y     and    y=bfz+f?, 

be  the  equation  of  the  perpendicular  PR,  and  because  it 
is  perpendicular  to  the  plane,  we  must  have  (by  the  last 
proposition) 

a'=^,     and     &'=:* 

Because  PQ  and  PR  are  two  lines  in  space,  if  we  des 
ignate  the  angle  included  by  F,  we  shall  have 


cos.  F=±  .  (prop<  8? 


But  the  cos.  F  is  the  same  as  the  sin.  PQR,  or  sin.  v, 
as  the  two  angles  are  complements  of  each  other. 

Making  this  change,  and  substituting  the  values  of  a' 
and  &',  we  have 


Bn., 


for  the  required  result. 

Cor.  —  When  #=0,  sin.  #=0,  and  this  hypothesis  gives 


for  the  equation  expressing  the  condition  that  the  given 
line  is  parallel  to  the  given  plane. 

"We  now  conclude  this  branch  of  our  subject  with  a 
few  practical  examples,  by  which  a  student  can  test  his 
knowledge  of  the  two  preceding  chapters. 


EQUATION    OF  A   PLANE.  269 

EXAMPLES. 

1.  —  What  is  the  distance  between  two  points  in  space  of 
which  the  co-ordinates  are 

3=3,  y=5,  *=—  2,  «'=—  2,  /=—  1,  *'=6. 

.An*.    11.180+. 

2.  —  (y  w/McA  the  co-ordinates  are 

x=I,  y=—  5,  z=—3,  3'=4,  y'=-4,  *'=!. 

Ans.     5^3  nearly. 

3.  —  TAe  equations  of  the  projections  of  a  straight  line  on  the 
co-ordinate  planes  (xz),  (yz),  are 

3=2*+l,  2/=Jz—  2, 

required  the  equation  of  projection  on  the  plane  (xy). 

Ans.    y—\x  —  2J. 

4.  —  7%e  equations  of  the  projections  of  a  line  on  the  co-ordi 
nate  planes  (xy)  and  (yz)  are 

2y—x  —  5        and        2y=z  —  4, 
required  the  equation  of  the  projection  on  the  plane  (xz). 

Ans.    x—z-\-1. 

5.  —  Required  the  equations  of  the   three  projections  of  a 
straight  line  which  passes  through  two  points  whose  co-ordinates 
are 

z/=2,  #'=1,  2'=0,  and  z"=—  3,  /=0,  z"=—  1. 
What  are  the  projections  on  the  planes  (xz)  and  (yz)  ? 

Ans.    x=5z+2,  y=z-\-\. 

And  from  these  equations  we  Und  the  projection  on  the 
plane  (xy\  that  is,  5?/=£-f  3. 
(See  Prop.  3,  Chap.  8.) 

6.  —  Required  the  angle  included  between  two  lines  whose 
equations  are 

I  of  the  1st,  and  x=z+2      I  of  the  2d. 


j  y=—z+I  j 

Ans.   F= 

(See  Prop.  8,  Chap.  8.) 
23* 


270  ANALYTICAL    GEOMETRY. 

7.  —  Find  the  angles  made  by  the  lines  designated  in  the  pre 
ceding  example,  with  the  co-ordinate  axes 

(See  Prop.  7,  Chap.  8.) 

(  36°  42'  with  JT,  f  54°44'  with  -J, 

Ans.  The  1st  line  1  57°  41'  20"    F,2d?line^  125°  16'        F, 

(74°  29'  54"    ^,  (54°  44'         Z. 

8.  —  Having  given  the  equation  of  two  straight  lines  in  space, 
as 


f  the  j      and 

;?/=— 

to  find  the  value  of  /3',  so  that  the  lines  shall  actually  intersect, 
and  to  find  the  co-ordinates  of  the  point  of  intersection. 


jx==< 


(See  Prop.  4,  Chap.  8.) 

9. — Given  the  equation  of  a  plane 

8x— Zy+z— 4=0, 

to  find  the  points  in  which  it  cuts  the  three  axes,  and  the  perpen 
dicular  distance  from  the  origin  to  the  plane. 

(Prop.  2.) 

Ans.  It  cuts  the  axis  of  X  at  the  distance  of  J  from 
the  origin ;  the  axis  of  Y  at  — 1 J ;  and  the  axis  of  Z  at 
+4. 

The  origin  is  .4649+  of  unity  below  the  plane. 

10. — Find  the  equations  for  the  intersections  of  the  two 
planes  (Prop.  4.) 

r-1-0, 


(On  the  plane  (xy)     17z— 10y+9=0. 

(  On  the  plane  (xz)     13x— 102+23=0. 
11. — Find  the  inclination  of  these  two  planes. 
(Prop.  6.) 

Ans.  41°  27'  41". 


EQUATION   OF  A  PLANE.  271 

12.  —  The  equations  of  a  line  in  space  are 

x=—2z+l,  and  y=32+2. 

find  the  inclination  of  this  Line  to  the  plane  represented  by 
the  equation  (Prop.  7.) 

Sx—  3y+z—  4=0. 

Ans.  48°  13'  13" 

13.  —  Find  the  angles  made  by  the  plane  whose  equation  is 

Sx—  Zy+z—  4=0, 
with  the  co-ordinate  planes. 

(Prop.  3.) 

f  83°  19'  27"  with  (xy\ 

Ans.{  110°  24'  38"  with  (xz). 

(  21°  34'  5"    with  (yz). 

14.  —  The  equation  of  a  plane  being 


Required  the  equation  of  a  parallel  plane  whose  perpendicu 
lar  distance  is  (a)  from  the  given  plane. 

Ans.  Because  the  planes  are  to  be  parallel,  their  equa 
tions  must  have  the  same  co-efficients,  J.,  B,  and  C. 

In  Prop.  2,  we  learn  that  the  perpendicular  distance  of 
the  origin  from  the  given  plane  may  be  represented  by 


Now,  as  the  planes  are  to  be  a  distance  a  asunder,  the 
distance  of  the  origin  from  the  required  plane  must  be 


or 


"Whence  the  equation  required  is 
Ax+By+ 


15.  —  Find  the  equation  of  the  plane  which  will  cut   the 
axis  of  Z  at  3,  the  axis  of  5  at  4,  and  the  axis  of  Y  at  5. 

Ans. 


272  ANALYTICAL   GEOMETRY. 

16. — Find  the  equation  of  the  plane  which  will  cut  the  axis 
of  X  at  3,  the  axis  of  Z  at  5,  and  which  will  pass  at  the 
perpendicular  distance  2  from  the  origin.  At  what  distance 
from  the  origin  will  this  plane  cut  the  axis  of  Y  ? 

Ans.     The  equation  of  the  plane  is 

10z+^89?/+62:— 30=0. 

30 
The  plane  cuts  the  axis  of  Y  at  ± . 

17. — Find  the  equations  of  the  intersection  of  the  two  planes 
whose  equations  are 

3x — 2j/ — z — 4=  0, 

The  equation  of  the  projection  of  the  inter 
section  on  the  plane  (xy)  is 

10z+?/— 6=0. 

A  ry\  Q 

23#— z— 16=0, 
and  that  on  the  plane  (yz)  is 

23^+10^+22=0. 

18. — Find  the  inclination  of  the  planes  whose  equations  are 
expressed  in  example  17. 

Ans.    7=60°  50'  55"  or  119°  9'  5". 

19. — A  plane  intersects  the  co-ordinate  plane  (xz)  at  an  in 
clination  of  50°,  and  the  co-ordinate  plane  (yz)  at  an  inclina 
tion  of  84°.  At  what  angle  will  this  plane  intersect  the  plane 

Ans.   F=40°38'6". 


MISCELLANEOUS   PROBLEMS.  273 


MISCELLANEOUS  PROBLEMS. 

1.  The  greatest  diameter  or  major  axis  of  an  ellipse  is 
40  feet,  and  a  line  drawn  from  the  center  making  an  an 
gle  of  36°  with  the  major  axis  and  terminating  in  the  el 
lipse  is  18  feet  long  ;  required  the  minor  axis  of  this  el 
lipse,  its  area  and  excentricity. 

NOTE.—  The  excentricity  of  an  ellipse  is  the  distance  of  either  focus 
from  the  center,  when  the  semi  major  axis  is  taken  as  unity. 

(  The  minor  axis  is  30.8752. 
AnsA  Area  of  the  ellipse,  969.972  sq.  feet. 
^  Excentricity  .63575. 

2.  If  equilateral  triangles  be  described  as  the  three  sides 
of  any  plane  triangle  and  the  centers  of  these  equilateral 
triangles  be  joined,  the  triangle  so  formed  will  be  equilat 
eral  ;  required  the  proof. 

Let  ABC  represent  any  plane 
triangle,  A,  B  and  C  denoting  the 
angles,  and  a,  b  and  c  the  respect 
ive  sides,  the  side  a  being  opposite 
the  angle  A,  and  so  on. 

On  A  C,  or  6,  suppose  an  equilat 
eral  triangle  to  be  drawn,  and  let 
P  be  its  center. 

Make  the  same  suppositions  in  regard  to  the  sides  c  and 
a,  finding  Pl  and  P2.  Draw  PPl  ,  P,  P2  and  PP2  ;  then 
is  PPl  P2  an  equilateral  triangle,  as  is  to  be  proved. 

We  shall  assume  the  principle,  which  may  be  easily 
demonstrated,  that  a  line  drawn  from  the  center  of  any  equi 
lateral  triangle  to  the  vertex  of  either  of  the  angles,  is  equal  to 

—  times  the  side  of  the  triangle.    Hence  we  have 


-      —  —  — 

\/3  \/3  V3  x/3 

Also,  the  angles  PJ.<7=30°,  PlVl_B=300, 


ANALYTICAL    GEOMETRY. 

and  so  on.  Now  it  is  obvious  that  the  angle  PAPi  is 
expressed  by  (J.-f  60°),  the  angle  P^P2  by  (5+60°),  and 
PCP2  by  (  (7+60°).  We  must  now  show  that  the  analyt 
ical  expressions  for  PPl  and  P1  P2  are  the  same.  In  an 
alytical  trigonometry  it  was  found  that  the  cosine  of  an  an 
gle,  A,  of  a  plane  triangle  would  be  given  by  the  equation 

cos.  A= 


Whence,         a2=62+c'2—  2bc  cos.  A. 

That  is,  The  square  of  one  side  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides,  minus  twice  the  rectangle  of  the 
other  two  sides  into  the  cosine  of  the  opposite  angle. 

Applying  this  to  the  triangle  PAPl  we  have 


Z  +3—  3"  cos' 
_  ,    c2  ,  a2    2ac 
Also,          PjP,,  =2+3  --  3~  cos-  (-#+60°)         (2) 

_  2    a     b2    2ab 
And  PP/=3+g  --  g-  cos.  ((7+60°)  (3) 

By  trigonometry,  cos.  (J.+60)=cos.  A  cos.  60  —  sin.  A 
sin.  60. 

But  cos.  60°=},  and  sin.  60=^3 

-v/3 

Whence,          cos.  (J.+60)=J  cos.  A  --  -  sin.  A 

2 

This  value  substituted  in  eq.  (1)  that  equation  becomes 
_  2    b2    c2    be  be 

PPi  K=3~+3  —  3  cos- 

b2+c2—a2  be 

But  cos.  A=  —  QA~  —  .    Whence  -o-cos.  A= 

This  value  of  ~  cos.  A  placed  in  eq.  (4),  gives 

_  2    262,2c2     b2    c2,a2,bc     . 
pp   —-TT-+—  -  —  -  —  -  +  -  +  —  sm.  A 
r*       6       6      6     6     6      v' 

Or, 


MISCELLANEOUS    PROBLEMS.  275 

By  a  like  operation  equation  (2)  becomes 


But  by  the  original  triangle  A  B  C  we  have 
sin.  J._sin.  B  a 

—^-       —  g—  >    or  sin-  A==b  sin.  B 

Placing  this  value  of  sin.  A  in  equation  (5)  that  equa 
tion  becomes 

—  ._aH-y+?  +*c_  Bin>  B  (7) 

1  6  v/8 

We  now  observe  that  the  second  members  of  (6)  and 
(7)  are  equal  ;  therefore,  PPl=P1  P2 

And  in  like  manner  we  can  prove  PPj=PP2.  There 
fore  the  triangle  PPtP2  has  been  shown  to  be  equilateral. 

PKOBLEM. 

G-iven,  the  excentricity  of  an  Ellipse,  to  find  the  difference 
between  the  mean  and  true  place  of  the  planet,  corres 
ponding  to  each  degree  of  the  mean  angle,  reckoned  from 
the  major  axis;  the  planet  describing  equal  sectors  or 
areas  in  equal  times,  about  one  of  the  foci,  the  center  of 
the  attractive  force. 

Let  AB  be  the  major  axis  of  an 
ellipse,  of  which  CB—  CA=A—\  is 
the  semi-transverse  axis,  and  also 
let  C  be  the  common  center  of  the 
ellipse  and  of  the  circle  of  which 
CB  is  the  radius.  Then  FC=e, 
and  F  is  the  focus  of  the  ellipse. 

Suppose  the  planet  to  be  at  _B, 
the  apogee  point  of  the  orbit,  (so  called  in  Astronomy). 
Also,  conceive  another  planet,  or  material  point,  to  be  at 
_B,  at  the  same  time.  Now,  the  planet  revolves  along  the 
ellipse,  describing  equal  areas  in  equal  times,  and  the  hy 
pothetical  planet  revolves  along  the  circle  BPQ,  describ- 


276  ANALYTICAL   GEOMETRY. 

ing,  in  equal  times,  equal  areas  and  equal  angles  about 
the  center  C. 

It  is  obvious  that  the  two  bodies  will  arrive  at  A  in  the 
same  time.  The  other  halves  of  the  orbits  will  also  be 
described  in  the  same  time,  and  the  two  bodies  will  be  to 
gether  again  at  the  point  B. 

But  at  no  other  points  save  at  A  and  at  B  (the  apogee 
and  perigee  points)  will  these  two  bodies  be  in  the  same 
line  as  seen  from  F,  and  the  difference  of  the  directions 
of  the  two  bodies  as  seen  from  the  focus  F  is  the  equation 
of  the  center.  For  instance,  suppose  the  planet  to  start 
from  B  and  describe  the  ellipse  as  far  as  p.  It  has  then 
described  the  area  BFp  of  the  ellipse,  about  the  focus  F. 
In  the  same  time  the  fictious  planet  in  the  circle  has  mov 
ed  along  the  circumference  BPto  Q,  describing  the  sector 
BCQ  about  the  center  C.  Now  the  areas  of  these  two 
sectors  must  be  to  each  other  as  the  area  of  the  ellipse  is 
to  the  area  of  the  circle.  That  is, 

sector  BFp  :  sector  BCQ  :  :  area  Ell.  :  area  Cir. 

Through  p  draw  PD  at  right  angles  to  J.J5,  and  repre 
sent  the  arc  of  the  circle  BP  by  x. 

Then     (7-D=cos.  x,  and  PD=sin.  x.    Draw  Op  and  CP. 

But,  denoting  the  semi-conjugate  axis  by  B,  we  have 
area  DpB  :  area  DPB  :  :  area  Ell.  :  area  Cir. 
:  :       B        :A 
::    pD        :  PD 

Also  we  have   ACpD  :  ACPD  :  :  pD  :  PD 

Hence,  area  DpB  :  ACpD  :  :  area  DPB  :  ACPD 

Therefore, 
area  DpB+A.CpD  :  area  DPB+ACPD  :  :  B  :  A 

or,      sector  CpB  :  sector  CPB  :  :  B:  A 

:  :  area  Ell.  :  area  Cir. 

Hence  it  follows  that 

sector  FpB  :  sector  CpB :  :  sector  CQB  :  sector  CPB 
"Whence 
sector  FpB—  sect.  CpB  :  sect.  CQB— sect.  CPB :  :B:A 


MISCELLANEOUS    PROBLEMS.  277 

or,  &FpC:  sector  QCP  :  :  B  :  A 

:  :  area  Ell.  :  area.  Cir. 

But  the  area  of  tlie  ellipse  is  xAB  and  the  area  of  the 
circle  is  A2x.     But  A=l  and  B=^\—  c2. 

The  area  of  the  triangle  FCp  is  \e  (pD\  and  the  area 
of  the  sector  is  Jy,    representing  the  arc  QP  by  y. 

Whence  JE(pD)  iyii  ^l^e^:  1.  (1) 

But  we  have        PD  :  pD  :  :  A  :  B 


:  :  1   :      1—  e2,       and  PD=sin.  x. 


Hence,    sin.  x  :  pD  :  :  1  :  ^1  —  e2;    pD=smx^l  —  e2 

This  value  of  pD  placed  in  (1)  that  proportion  becomes 
e  sin.  x^l  —  e2  :  y  :  :  ^1  —  e2  :  1 

Or,  e  sin.  x  :  y  :  :  1  :  1.      y=e  sin.  x.  (2) 

DEFINITIONS.  —  1st.  The  angle  x,  in  astronomy,  is  called 
the  excentric  anomaly. 

2d.  The  angle  QCB,  or  (x+y)  is  called  the  mean 
anomaly. 

3d.  The  angle  p  FB  is  called  the  true  anomaly. 

4th.  The  difference  between  QCB  or  nCB  (of  the  tri 
angle  FnC)  and  nFC  (which  is  the  angle  n  of  the  trian 
gle  CFn)  is  the  equation  of  the  center. 

The  angle  QCB,  the  mean  anomaly,  is  an  angle  at  the 
center  of  the  ellipse,  which  is  equal  to  the  sum  of  the  an 
gles  at  n  and  F;  that  is,  n  taken  from  the  angle  at  the 
center  will  give  the  true  angle  at  the  focus,  F. 

"We  will  designate  the  angle  pFB  by  t.  Now,  by  the 
polar  equation  of  an  ellipse,  we  have 


Again,  by  the  triangle  FDp,  we  find, 


But  _    ~FD2=(e+coa.  x)2=e2+2e  cos.  z-f  cos.2z 
And  pJD  =sin.2  x  (1  —  e2)=sin.2  x  —  e  2  sin.2  x 
Therefore,  FD2+pD2=e2+2e  cos.  x+l—e2sm.2x 
But  e2  sin.2  x—e,  2 

2.4 


278  ANALYTICAL   GEOMETBY. 

Substituting  this  value  of  &  sin.2-£  in  the  preceding 
expression  we  have 


cos. 

"Whence         Fp=</  FD2+pDz=l+e  cos.  x. 
Equaling  these  two  values  of  Fp  and  we  obtain 
1  —  e2=(l-j-c  cos.  x)  (1  —  e  cos.  t) 

e-f-cos.  x 
Whence         cos.  ^—- 


Here  we  have  a  value  of  t  in  terms  of  x  and  e,  but  the 
equation  is  not  adapted  to  the  use  of  logarithms. 
By  equation  (27)  Plane  Trigonometry,  we  have 
1  —  cos.  t 


If  the  value  of  cos.  t  from  equation  (3)  be  placed  in  this 
we  shall  have 

e+cos.  x 


..  _ 


1+e  cos.  x        i_|_g  Cos.  x— e— cos.  x 


1-fecos.a; 
Or 


e+coa.x         l+ecos.z+e-t    os.z 


(l+e)+(l+6)  cos.  a;      (l+c)  (1+cos.z) 
That  is,    tan.  J^=  (  l~e  )  *  tan.  Jx.  (4) 


From  eq.  (2)  we  obtain 

.Mean  Anomaly  =£-{-£  sin.  x.  (5) 

By  assuming  a:,  equation  (5)  gives  the  Mean  Anomaly. 
Then  equation  (4)  gives  the  corresponding  True  Anomaly. 
To  apply  these  equations  to  the  apparent  solar  orbit,  the 
value  of  e  is  .0167751  the  radius  of  the  circle  being  unity. 
But  y=e  sin.  x,  and  as  y  is  a  portion  of  the  circumfer 
ence  to  the  radius  unity,  we  must  express  e  in  some 
known  part  of  the  circumference,  one  degree,  for  exam 
ple,  as  the  unit. 

Because  180°  is  equal  to  3.14159265,  therefore  the  value 
of  e,  in  degrees,  is  found  by  the  following  proportion. 


MISCELLANEOUS   PROBLEMS.  £79 

3.14159265  :  180°  :  :  .0167751  :  d  degrees. 
By  log.,  log.  0167751    —2.2246652 

log.  180°  2.2552725 

0.4799377 

log.  n  0.4971499 

Log.  e,  in  degrees,  of  arc,       —  1,9827878 

Add  log.  60         1.7781513 

Log.  e,  in  min.  of  arc,          1.7609391  constant  log. 

lZ7  /  0.9832249 

-1-992714  cons.  log. 


"We  are  now  prepared  to  make  an  application  of  equa 
tions  (4)  and  (5) 

For  example,  we  require  the  equation  of  the  center 
for  the  solar  orbit,  corresponding  to  28°  of  mean  anom 
aly,  reckoning  from  the  apogee.  The  excentric  anomaly 
is  less  than  the  mean  by  about  half  of  the  value  of  the 
equation  of  the  center  at  any  point;  and  x  must  be  as 
sumed. 

Thus,      suppose  z=27°  32'  ;  then  Jz=13°  46' 
sin.  z=sin.  27°  32'  9.664891 

Constant,  1.760939 

e  sin.  x=         26'  6518     1.425830 
Add  x      27°  32' 


Mean  Anomaly=27°  58'  39"! 

Tan.  Jx  13°  46'        9.389178 
Const.    —1.992714 


tan.  J*  13°  32'  59"        9.381892 

2 


True  anomaly  27°    5'  58" 
Mean  Anomaly  27°  58'  39"! 

Equation  of  center         52r  41"!        corresponding  to  the 
mean  anomaly  of  27°  58'  39"1,  not  to  28°  as  was  required. 


280  ANALYTICAL    GEOMETRY. 

Now  let  us  take  z=27°  40';  then  ^=13°  50' 
sin.  x    27°  40  9.666824 

Con.  1.760939 


e  sin.  x        26'  777        1.427763 
Add  x  27°  40' 


Mean  Anomaly,  28°    6'  46"6 

tan.  }z=13°  50'        9.391360 
Con.     —1.992714 


tan.  \t    13°  36'  43"        9.384074 
2 


*=27°  13'  26" 
Mean  anomaly  28°    6'  46"  6 

Eq.  center,  53'  20  "6 

corresponding  to  28°  6'  46"6. 

Now,  we  can  find  the  equation  corresponding  to  28°  by 
the  following  obvious  proportion : 

28°    6'  46"6        53'  20"6        28°  00'  00" 
27    58   39  1        52  41  1        27      5  39  1 


8'    7"5  :  39"5  :  :  V  20"9  :  4"7 

Add  52'  41"! 


Equation  or  value  sought,  52'  45"! 

In  like  manner  we  can  find  the  value  of  the  equation 
of  the  center  of  any  and  every  other  degree  of  the  mean 
anomaly  in  the  orbit  of  the  sun,  or  any  other  orbit,  when 
the  excentricity  is  known. 


LOGARITHMIC  TABLES 


ALSO    A    TABLE    OF 


NATURAL     AND     LOGARITHMIC 


SINES,  COSINES,  AND  TANGENTS, 


TO    EVERY    MINUTE    OF    THE    QUADRANT. 


LOGARITHMS    OF    NUMBERS 

FROM 

1    TO    10000, 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

0  000000 

26 

414973 

51 

1  707570 

76 

1  880814 

0 

0  301030 

27 

431364 

52 

1  716003 

77 

1  886491 

3 

0  477121 

28 

447158 

53 

1  724276 

78 

1  892095 

4 

0  6020GO 

29 

462398 

54 

1  732394 

79 

1  897627 

5 

0  698970 

30 

477121 

55 

1  740363 

80 

1  903090 

6 

0  778151 

31 

491362 

56 

1  748188 

81 

1  908485 

7 

0  845098 

32 

505150 

57 

1  755875 

82 

1  913814 

8 

0  903090 

33 

6J8514 

58 

1  763428 

83 

1  919078 

9 

0  954243 

34 

531479 

59 

1  770852 

84 

1  924279 

10 

1  000000 

35 

544068 

60 

1  778151 

85 

1  929419 

11 

041393 

36 

556303 

61 

1  785330 

86 

1  934498 

12 

079181 

37 

568202 

62 

1  792392 

87 

1  939519 

13 

113943 

38 

679784 

63 

1  799341 

88 

1  944483 

14 

146128 

39 

1  591065 

64 

1  808180 

89 

1  949390 

15 

176091 

40 

1  602060 

65 

1  812913 

90 

1  954243 

16 

204120 

41 

1  612784 

66 

1  819544 

91 

1  959041 

17 

230449 

42 

1  623249 

67 

1  826075 

92 

1  963788 

18 

255273 

43 

1  633468 

68 

832509 

93 

1  968483 

19 

278754 

44 

1  643453 

69 

838849 

94 

1  973128 

20 

301030 

45 

1  653213 

70 

845098 

95 

1  977724 

21 

1  322219 

46 

662578 

71 

851258 

96 

1  982271 

22 

1  342423 

47 

672098 

72 

857333 

97 

1  986772 

23 

1  361728 

48 

681241 

73 

863323 

98 

1  991226 

24 

1  380211 

49 

690196 

74 

869232 

99 

1  995635 

25 

1  397940 

50 

698970 

75 

8750ol 

100 

2  000000 

NOTE.  In  the  following  table,  in  the  last  nine  columns  of  each  page,  where 

the  first  or  leading  figures  change  from  9's  to  O's,  points  or  dots  are  now 

introduced  instead  of  the  O's  through  the  rest  of  the  line,  to  catch  the  eye, 

and  to  indicate  that  from  thence  the  corresponding  natural  number  in 

the  first  column  stands  in  the  next  lower  line,  and  its  annexed  first  two 

figures  of  the  Logarithms  in  the  second  column. 

LOGARITHMS  OF  NUMBERS.      3 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

000000 

0434 

0868 

1301 

1734 

2166 

2598 

3029 

3461 

3891 

101 

4321 

4750 

5181 

5609 

6038 

6466 

6894 

7321 

7748 

8174 

102 

8'JOO 

9026 

9451 

9876 

.300 

.724 

1147 

1570 

1993 

2416 

103 

01-2.S37 

3259 

3680 

4100 

4521 

4940 

5360 

5779 

6197 

6616 

104 

7033 

7451 

7868 

8284 

8700 

9116 

9532 

9947 

.361 

.775 

105 

021  1S9 

1603 

2016 

2428 

2841 

3252 

3664 

4075 

4486 

4896 

103 

530'J 

5715 

6125 

6533 

6942 

7350 

7757 

8164 

8571 

8978 

107 

9384 

9789 

.195 

.600 

1004 

1408 

1812 

2216 

2619 

3021 

108 

033424 

3826 

4227 

4628 

5029 

5430 

5830 

6230 

6629 

7028 

109 

7426 

7825 

8223 

8620 

9017 

9414 

9811 

.207 

.602 

.998 

110 

041393 

1787 

2182 

2676 

2969 

3362 

3755 

4148 

4540 

4932 

111 

5323 

5714 

6105 

6495 

6885 

7275 

7664 

8053 

8442 

8830 

112 

9218 

9606 

9993 

.380 

.766 

1153 

1538 

1924 

2309 

2694 

113 

053078 

3463 

3846 

4230 

4613 

4996 

5378 

5760 

6142 

6524 

114 

6905 

7286 

7666 

8046 

8426 

8805 

9185 

9563 

9942 

.320 

115 

050898 

1075 

1452 

1829 

2206 

2582 

2958 

3333 

3709 

4083 

116 

4458 

4832 

5206 

5580 

5953 

6326 

6699 

7071 

7443 

7815 

117 

8186 

8557 

8928 

9298 

9668 

..38 

.407 

.776 

1145 

1514 

118 

071882 

2250 

2617 

2985 

3352 

3718 

4085 

4451 

4816 

6182 

119 

5547 

6912 

6276 

6640 

7004 

7368 

7731 

8094 

8457 

8819 

120 

9181 

9543 

9904 

.266 

.626 

.987 

1347 

1707 

2067 

2426 

121 

082785 

3144 

3503 

3861 

4219 

4576 

4934 

5291 

5647 

6004 

122 

6360 

6716 

7071 

7426 

7781 

8136 

8490 

8845 

9198 

9552 

123 

9905 

.258 

.611 

.963 

1315 

1667 

2018 

2370 

2721 

3071 

124 

093422 

3772 

4122 

4471 

4820 

5169 

5518 

5866 

6215 

6562 

125 

6910 

7257 

7604 

7951 

8298 

8644 

8990 

9335 

9681 

1026 

136 

100371 

0715 

1059 

1403 

1747 

2091 

2434 

2777 

3119 

3462 

127 

3804 

4146 

4487 

4828 

5169 

5510 

5851 

6191 

6531 

6871 

128 

7210 

7549 

7888 

8227 

8565 

8903 

9241 

9579 

9916 

.253 

129 

110590 

0926 

1263 

1599 

1934 

2270 

2605 

2940 

3275 

3609 

130 

3S43 

4277 

4611 

4944 

5278 

6611 

5943 

6276 

6608 

6940 

131 

7271 

7603 

7934 

82G5 

8595 

8926 

9256 

9586 

9915 

0245 

132 

120574 

0903 

1231 

1560 

1888 

2216 

2544 

2871 

3198 

3525 

133 

3852 

4178 

4504 

4830 

5156 

5481 

5806 

6131 

6456 

6781 

134 

7105 

7429 

7753 

8076 

8399 

8722 

9045 

9368 

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135 

130334 

0655 

0977 

1298 

1619 

1939 

2260 

2580 

2900 

3219 

136 

3539 

3858 

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4496 

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5769 

6086 

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137 

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7987 

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138 

9879 

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1136 

1450 

1763 

2076 

2389 

2702 

139 

143015 

3327 

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140 

6128 

6438 

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141 

9219 

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1063 

1370 

1676 

1982 

142 

152288 

2594 

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3815 

4120 

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4728 

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143 

5336 

5640 

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7457 

7759 

8061 

144 

8362 

8G64 

8965 

9266 

9567 

9868 

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.769 

10G8 

145 

161308 

1667 

1967 

2266 

2564 

2863 

3161 

3460 

3758 

4055 

146 

4353 

4650 

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5541 

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147 

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9086 

9380 

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148 

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0555 

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2019 

2311 

2603 

2895 

149 

3186 

3478 

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4060 

4351 

4641 

4932 

5222 

5512 

5802 

18 


4               LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

150 

176091 

6381 

6670 

6959 

7248 

7536 

7825 

8113 

8401 

8689 

151 

8977 

9264 

9552 

9839 

.126 

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.699 

.985 

1272 

1558 

152 

181844 

2129 

2415 

2700 

2985 

3270 

3555 

3839 

4123 

4407 

153 

4691 

4975 

5259 

5542 

5825 

6108 

6391 

6674 

6956 

7239 

154 

7521 

7803 

8084 

8366 

8647 

8928 

9209 

9490 

9771 

..51 

281 

155 

190332 

0612 

0892 

1171 

1451 

1730 

2010 

2289 

2567 

2846 

156 

3125 

3403 

3681 

3959 

4237 

4514 

4792 

5069 

5346 

5623 

157 

5899 

6176 

6453 

6729 

7005 

7281 

7556 

7832 

8107 

8382 

158 

8657 

8932 

9206 

9481 

9755 

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.303 

.577 

.850 

1124 

159 

201397 

1670 

1943 

2216 

2488 

2761 

3033 

3305 

3577 

3848 

273 

160 

4120 

4391 

4663 

4934 

5204 

5475 

5746 

6016 

6286 

6556 

161 

6826 

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7634 

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8710 

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9247 

162 

9515 

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1121 

1388 

1654 

1921 

163 

212188 

2454 

2720 

2986 

3252 

3518 

3783 

4049 

4314 

4579 

164 

4844 

5109 

5373 

5638 

5902 

6166 

6430 

6694 

6957 

7221 

264 

165 

7484 

7747 

8010 

8273 

8536 

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166 

220108 

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0631 

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1153 

1414 

1675 

1936 

2196 

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167 

2716 

2976 

3236 

3496 

3755 

4015 

4274 

4533 

4792 

5051 

168 

6309 

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6084 

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6600 

6858 

7115 

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7630 

169 

7887 

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8657 

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9938 

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257 

170 

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0704 

0960 

1215 

1470 

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177 

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178 

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2125 

2368 

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179 

2853 

3096 

3338 

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3822 

4064 

4306 

4548 

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5031 

242 

180 

5273 

5514 

5755 

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6237 

6477 

6718 

6958 

7198 

7439 

181 

7679 

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8637 

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182 

260071 

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1263 

1501 

1739 

1976 

2214 

183 

2451 

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3162 

3399 

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4109 

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4582 

184 

4818 

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6232 

6467 

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235 

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186 

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189 

6462 

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229 

190 

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9211 

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191 

281033 

1261 

1488 

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2169 

2396 

2622 

2849 

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192 

3301 

3527 

3753 

3979 

4205 

4431 

4656 

4882 

5107 

5332 

193 

5557 

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6681 

6905 

7130 

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194 

7802 

8026 

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9366 

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224 

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2256 

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7104 

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8416 

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199 

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OFNUMBERS.               5 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

200 

301030 

1247 

1464 

1681 

1898 

2114 

2331 

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2764 

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201 

3196 

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3628 

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4059 

4275 

4491 

4706 

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5136 

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7038 

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203 

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204 

9630 

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1118 

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1542 

212 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3023 

3234 

3445 

3656 

206 

3867 

4078 

4289 

4499 

4710 

4920 

5130 

5340 

5551 

5760 

207 

5970 

6180 

6390 

6599 

6809 

7018 

7227 

7436 

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208 

8083 

8272 

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8689 

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9106 

9314 

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209 

320146 

0354 

0562 

0769 

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1184 

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2012 

207 

210 

2219 

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3458 

3685 

3871 

4077 

211 

4282 

4488 

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5721 

5926 

6131 

212 

6336 

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6745 

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7155 

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7563 

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213 

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215 

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3649 

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4253 

216 

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217 

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233 

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185 

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1253 

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2175 

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4015 

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238 

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244 

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178 

245 

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1112 

1288 

1464 

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1993 

2169 

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247 

2697 

2873 

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248 

4452 

4627 

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5326 

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5850  6025 

249 

6199 

6374 

6548 

6722 

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7071 

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6               LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

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8981 

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251 

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1056 

1228 

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401401 

1573 

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2261 

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2605 

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253 

3121 

3292 

3464 

3635 

3807 

3978 

4149 

4320 

4492 

4663 

254 

4834 

5005 

5176 

5346 

5517 

5688 

5858 

6029 

6199 

6370 

171 

255 

6540 

6710 

6881 

7051 

7221 

7391 

7561 

7731 

7901 

8070 

256 

8240 

8410 

8579 

8749 

8918 

9087 

9257 

9426 

9595 

9764 

257 

9933 

.102 

.271 

.440 

.609 

.777 

.946 

1114 

1283 

1451 

258 

411620 

1788 

1956 

2124 

2293 

2461 

2629 

2796 

2964 

3132 

259 

3300 

3467 

3635 

3803 

3970 

4137 

4305 

4472 

4639 

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260 

4973 

5140 

5307 

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5641 

5808 

5974 

6141 

6308 

6474 

261 

6641 

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7139 

1303 

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7638 

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262 

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263 

9956 

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1110 

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1439 

264 

421604 

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2261 

2426 

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2754 

2918 

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265 

3246 

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3574 

3737 

3901 

4065 

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4555 

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266 

4882 

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5208 

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5697 

6860 

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6186 

6349 

267 

6511 

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6836 

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7161 

7324 

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7648 

7811 

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268 

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8297 

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9106 

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9429 

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269 

9752 

9914 

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.398 

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1042 

1203 

270 

431364 

1525 

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2007 

2167 

2328 

2488 

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2809 

271 

2969 

3130 

3-290 

3450 

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3930 

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4-249 

4409 

272 

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7433 

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274 

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1276 

7428 

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287 

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288 

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289 

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1499 

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1948 

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2248 

290 

2398 

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2847 

2997 

3146 

3296 

3445 

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3744 

291 

0893 

4042 

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4340 

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4639 

4788 

4936 

5085 

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292 

5383 

5532 

5680 

5829 

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6126 

6274 

6423 

6571 

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293 

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7016 

7164 

7312 

7460 

7608 

7766 

7904 

8052 

8200 

294 

8347 

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9380 

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147 

295 

9822 

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296 

471292 

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2171 

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2464 

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297 

2756 

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3925 

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298 

4216 

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5235 

5381 

5526 

299 

6671 

5816 

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6687 

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!  

l 

OF  NUMBERS.              7 

N. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

300 

477121 

7266 

7411 

7555 

7700 

7844 

7989 

8133 

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301 

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9143 

9287 

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9575 

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302 

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1586 

1729 

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2159 

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2874 

3016 

3159 

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3730 

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4015 

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142 

305 

4300 

4442 

4585 

4727 

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5011 

6153 

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5437 

5579 

306 

5721 

5863 

6005 

6147 

6289 

6430 

6572 

6714 

6855 

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307 

7138 

7280 

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7563 

7704 

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8127 

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308 

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309 

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310 

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311 

2760 

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312 

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315 

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318 

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6306 

6432 

344 

6558 

6685 

6811 

6937 

7060 

7189 

7315 

7441 

7567 

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129 

345 

7819 

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347 

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1080 

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2203 

2327 

2452 

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2701 

349 

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3199 

3323 

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3696 

3820 

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0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

350 

544068 

4192 

4316 

4440 

4564 

4688 

4812 

4936 

5060 

5183 

351 

5307 

5431 

5555 

5578 

5805 

5925 

6049 

6172 

6296 

6419 

352 

6543 

6666 

6789 

6913 

7036 

7159 

7282 

7405 

7529 

7652 

353 

7775 

7898 

8021 

8144 

8267 

8389 

8512 

8635 

8758 

8881 

354 

9003 

9126 

9249 

9371 

9494 

9616 

9739 

9861 

9984 

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122 

355 

550228 

0351 

0473 

0595 

0717 

0840 

0962 

1084 

1206 

1328 

356 

1450 

1572 

1694 

1816 

1938 

2060 

2181 

2303 

2425 

2547 

357 

2668 

2790 

2911 

3033 

3155 

3276 

3393 

3519 

3640 

3762 

358 

3883 

4004 

4126 

4247 

4368 

4489 

4610 

4731 

4852 

4973 

359 

5094 

5215 

5346 

5457 

5578 

5699 

5820 

5940 

6061 

6182 

360 

6303 

6423 

6544 

6664 

6785 

6905 

7026 

7146 

7267 

7387 

361 

7507 

7627 

7748 

7868 

7988 

8108 

8228 

8349 

8469 

8589 

362 

8709 

8829 

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9308 

9428 

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9667 

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363 

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364 

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1340 

1459 

1578 

1698 

1817 

1936 

2056 

2173 

365 

2293 

2412 

2531 

2650 

2769 

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3006 

3125 

3244 

3362 

366 

3481 

3600 

3718 

3837 

3955 

4074 

4192 

4311 

4429 

4548 

367 

4666 

4784 

4903 

5021 

5139 

5257 

5376 

5494 

5612 

5730 

368 

5848 

5966 

6084 

6202 

6320 

6437 

6555 

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6791 

6909 

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7026 

7144 

7262 

7379 

7497 

7614 

7732 

7849 

7967 

8084 

370 

8202 

8319 

8436 

8554 

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8788 

8905 

9023 

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371 

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9725 

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9959 

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372 

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0893 

1010 

1126 

1243 

1359 

1476 

1592 

373 

1709 

1825 

1942 

2058 

2174 

2291 

2407 

2522 

2639 

2755 

374 

2872 

2988 

3104 

3220 

3336 

3452 

3568 

3634 

3800 

2915 

116 

375 

4031 

4147 

4263 

4379 

4494 

4610 

4726 

4841 

4957 

5072 

376 

5188 

5303 

5419 

5534 

5650 

5765 

5880 

5996 

6111 

6226 

377 

63  il 

6457 

6572 

6687 

6802 

6917 

7032 

7147 

7262 

7377 

378 

7492 

7607 

7722 

7836 

7951 

8066 

8181 

8295 

8410 

8525 

379 

8639 

8754 

8868 

8983 

9097 

9212 

9326 

9441 

9555 

9669 

380 

9784 

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.811 

381 

580925 

1039 

1153 

1267 

1381 

1495 

1608 

1722 

1836 

1950 

382 

20G3 

2177 

2291 

2404 

2518 

2631 

2745 

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2972 

3085 

383 

3199 

3312 

3426 

3539 

3652 

3765 

3879 

3992 

4105 

4218 

384 

4331 

4444 

4557 

4670 

4783 

4896 

5009 

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5235 

5348 

385 

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5574 

5686 

5799 

5912 

6024 

6137 

0250 

6362 

6475 

386 

6587 

6700 

6812 

6925 

7037 

7149 

7262 

7374 

7486 

7599 

387 

7711 

7823 

7935 

8047 

8160 

8272 

8384 

8496 

8608 

8720 

388 

8832 

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9056 

9167 

9279 

9391 

9503 

9615 

9726 

9834 

389 

9950 

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390 

591065 

1176 

1287 

1399 

1510 

1621 

1732 

1843 

1955 

2066 

391 

2177 

2288 

2399 

2510 

2621 

2732 

2843 

2954 

3064 

3175 

392 

3286 

3397 

3508 

3618 

3729 

3840 

3950 

4061 

4171 

4282 

393 

4393 

4503 

4614 

4724 

4834 

4945 

5055 

5165 

5276 

5386 

394 

5496 

5606 

5717 

5827 

5937 

6047 

6157 

6267 

6377 

6487 

110 

395 

6597 

6707 

6817 

6927 

7037 

7146 

7256 

7366 

7476 

7586 

396 

7695 

7805 

7914 

8024 

8134 

8243 

8353 

8462 

8572 

8681 

397 

8791 

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9009 

9119 

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5,656 

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9774 

398 

9883 

9992 

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.537 

.646 

.755 

.864 

399 

600973 

1082 

1191 

1299 

1408 

1517 

1625 

1734 

1843 

1961 

OFNUMBERS.               9 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

400 

602060 

2169 

2277 

2386 

2494 

2603 

2711 

2819 

2928 

3036 

401 

3144 

3253 

3361 

3469 

3573 

3686 

3794 

3902 

4010 

4118 

402 

4226 

4334 

4442 

4550 

4658 

4766 

4874 

4982 

6089 

5197 

403 

5305 

5413 

5521 

5628 

6736 

5844 

5951 

6059 

6166 

6274 

404 

6381 

6489 

6596 

6704 

6811 

6919 

7026 

7133 

7241 

7348 

108 

405 

7455 

7562 

7669 

7777 

7884 

7991 

8098 

8205 

8312 

8419 

406 

8526 

8633 

8740 

8847 

8954 

9061 

9167 

9274 

9381 

9488 

407 

9594 

9701 

9808 

9914 

..21 

.128 

.234 

.341 

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.654 

408 

610660 

0767 

0873 

0979 

1086 

1192 

1298 

1405 

1511 

1617 

409 

1723 

1829 

1936 

2042 

2148 

2254 

2360 

2466 

2572 

2678 

410 

2784 

2890 

2996 

3102 

3207 

3313 

3419 

3525 

3630 

3736 

411 

3842 

3947 

4053 

4159 

4264 

4370 

4475 

4581 

4686 

4792 

412 

4897 

5003 

5108 

5213 

5319 

5424 

6529 

6634 

5740 

5845 

413 

5950 

6055 

6160 

6265 

6370 

6476 

6581 

6686 

6790 

6895 

414 

7000 

7105 

7210 

7315 

7420 

7526 

7629 

7734 

7839 

7943 

415 

8048 

8153 

8257 

8362 

8466 

8571 

8676 

8780 

8884 

8989 

416 

9293 

9198 

9302 

9408 

9511 

9615 

9719 

9824 

9928 

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417 

620136 

0240 

0344 

0448 

0552 

0656 

0760 

0864 

0968 

1072 

418 

1176 

1280 

1384 

1488 

1592 

1695 

1799 

1903 

2007 

2110 

419 

2214 

2318 

2421 

2525 

2628 

2732 

2835 

2939 

3042 

3146 

420 

3249 

3353 

3456 

3559 

3663 

3766 

3869 

3973 

4076 

4179 

421 

4282 

4385 

4488 

4591 

4695 

4798 

4901 

6004 

610? 

5210 

422 

5312 

5415 

5518 

5621 

6724 

5827 

5929 

6032 

6135 

6238 

423 

6340 

6443 

6546 

6648 

6751 

6853 

6956 

7058 

7161 

7263 

424 

7366 

7468 

7571 

7673 

7775 

7878 

7980 

8082 

8185 

8287 

103 

425 

8389 

8491 

8593 

8695 

8797 

8900 

9002 

9104 

9206 

9308 

426 

9410 

9512 

9613 

9715 

9817 

9919 

.  .21 

.123 

.224 

.326 

427 

630428 

0530 

0631 

0733 

0835 

0936 

1038 

1139 

1241 

1342 

428 

1444 

1545 

1647 

1748 

1849 

1951 

2052 

2153 

2255 

2356 

429 

2457 

2559 

2660 

2761 

2862 

2963 

3064 

3165 

3266 

3367 

430 

3468 

3569 

3670 

3771 

3872 

3973 

4074 

4175 

4276 

4376 

431 

4477 

4578 

4679 

4779 

4880 

4981 

5081 

5182 

5283 

5383 

432 

5484 

5584 

5685 

5785 

6886 

6986 

6087 

6187 

6287 

6388 

433 

6488 

6588 

6688 

6789 

6889 

6989 

7089 

7189 

7290 

7390 

434 

7490 

7590 

7690 

7700 

7890 

7990 

8090 

8190 

8290 

8389 

435 

8489 

8589 

8689 

8789 

8888 

8988 

9088 

9188 

9287 

9387 

436 

9486 

9586 

9686 

9785 

9885 

9984 

..84 

.183 

.283 

.382 

437 

640481 

0581 

0680 

0779 

0879 

0978 

1077 

1177 

1276 

1375 

438 

1474 

1573 

1672 

1771 

1871 

1970 

2069 

2168 

2267 

2366 

439 

2465 

2563 

2662 

2761 

2860 

2959 

3058 

3156 

3255 

3354 

440 

3453 

3551 

3650 

3749 

3847 

3946 

4044 

4143 

4242 

4340 

441 

4439 

4537 

4636 

4734 

4832 

4931 

5029 

5127 

6226 

5324 

442 

5422 

5521 

5619 

5717 

6815 

6913 

6011 

6110 

6208 

6306 

443 

6404 

6502 

6600 

6698 

6796 

6894 

6992 

7039 

7187 

7285 

444 

7383 

7481 

7579 

7676 

7774 

7872 

7969 

8067 

8165 

8262 

93 

445 

8360 

8458 

8555 

8653 

8750 

8848 

8945 

9043 

9140 

9237 

446 

9335 

9432 

9530 

9627 

9724 

9821 

9919 

.  .16 

.113 

.210 

447 

650308 

0405 

0502 

0599 

0696 

0793 

0890 

0987 

1084 

1181 

448 

1278 

1375 

1472 

1569 

1666 

1762 

1859 

1956 

2053 

2150 

449 

2246 

2343 

2440 

2530 

2633 

2730 

2826 

2923 

3019 

3116 

10              LOGARITHMS 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

653213 

3309 

3405 

3502 

3598 

3695 

3791 

3888 

3984 

4080 

451 

4177 

4273 

4369 

4465 

4562 

4658 

4754 

4850 

4946 

5042 

452 

5138 

5235 

5331 

5427 

5526 

5619 

5715 

5810 

5906 

6002 

453 

6098 

6194 

6290 

6386 

6482 

6577 

6673 

6769 

6864 

6960 

454 

7056 

7152 

7247 

7343 

7438 

7534 

7629 

7725 

7820 

7916 

96 

455 

8011 

8107 

8202 

8298 

8393 

8488 

8584 

8679 

8774 

8370 

456 

8965 

9060 

9155 

9250 

9346 

9441 

9536 

9631 

9726 

9821 

457 

9916 

.  .11 

.106 

.201 

.296 

.391 

.486 

.681 

.676 

.771 

458 

660865 

0960 

1055 

1150 

1245 

1339 

1434 

1529 

1623 

1718 

459 

1813 

1907 

2002 

2096 

2191 

2286 

2380 

2475 

2569 

2663 

460 

2758 

2852 

2947 

3041 

3135 

3230 

3324 

3418 

3512 

3607 

461 

3701 

3795 

3889 

3983 

4078 

4172 

4266 

4360 

4454 

4548 

462 

4642 

4736 

4830 

4924 

5018 

5112 

6206 

5299 

5393 

6487 

463 

5581 

5675 

5769 

5862 

5956 

6050 

6143 

6237 

6331 

6424 

464 

6518 

6612 

6705 

6799 

6892 

6986 

7079 

7173 

7266 

7360 

465 

7453 

7546 

7640 

7733 

7826 

7920 

8013 

8106 

8199 

8293 

466 

8386 

8479 

8572 

8665 

8759 

8852 

8945 

9038 

9131 

9324 

467 

9317 

9410 

9503 

9596 

9689 

9782 

9875 

9967 

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.153 

468 

670241 

0339 

0431 

0524 

0617 

0710 

0802 

0895 

0988 

1080 

469 

1173 

1265 

1358 

1451 

1543 

1636 

1728 

1821 

1913 

2005 

470 

2098 

2190 

2283 

2375 

2467 

2560 

2652 

2744 

2836 

2929 

471 

3021 

3113 

3205 

3297 

3390 

3482 

3574 

3666 

3758 

3850 

472 

3942 

4034 

4126 

4218 

4310 

4402 

4494 

4586 

4677 

4769 

473 

4861 

4953 

5045 

5137 

5228 

5320 

5412 

6503 

5595 

6687 

474 

6778 

5870 

5962 

6053 

6145 

6236 

6328 

6419 

6511 

6602 

91 

4€5 

6694 

6785 

6876 

6968 

7059 

7151 

7242 

7333 

7424 

7616 

476 

7607 

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7789 

7881 

7972 

8063 

8154 

8245 

8336 

8427 

477 

8518 

8609 

8700 

8791 

8882 

8972 

9064 

9155 

9246 

9337 

478 

9428 

9519 

9610 

9700 

9791 

9882 

9973 

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.154 

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479 

680336 

0426 

0517 

0607 

0698 

0789 

0879 

0970 

1060 

1151 

480 

1241 

1332 

1422 

1513 

1603 

1693 

1784 

1874 

1964 

2055 

481 

2145 

2235 

2326 

2416 

2506 

2696 

2686 

2777 

2867 

2957 

482 

3047 

3137 

3227 

3317 

3407 

3497 

3587 

3677 

3767 

3857 

483 

3947 

4037 

4127 

4217 

4307 

4396 

4486 

4576 

4666 

4756 

484 

4854 

4935 

5025 

5114 

5204 

5294 

6383 

5473 

5563 

5652 

485 

5742 

5831 

5921 

6010 

6100 

6189 

6279 

6368 

6458 

6547 

486 

6636 

6726 

6815 

6904 

6994 

7083 

7172 

7261 

7351 

7440 

487 

7529 

7618 

7707 

7796 

7886 

7975 

8064 

8153 

8242 

8331 

488 

8420 

8509 

8598 

8687 

8776 

8865 

8953 

9042 

9131 

9220 

489 

9309 

9398 

9486 

9575 

9664 

9763 

9841 

9930 

..19 

.107 

490 

690196 

0285 

0373 

0362 

0550 

0639 

0728 

0816 

0905 

0993 

491 

1081 

1170 

1258 

1347 

1435 

1524 

1612 

1700 

1789 

1877 

492 

Ib65 

2053 

2142 

2230 

2318 

2406 

2494 

2583 

2671 

2759 

493 

2847 

2935 

3023 

3111 

3199 

3287 

3375 

3463 

3651 

3639 

494 

3727 

3815 

3903 

3991 

4078 

OQ 

4166 

4254 

4342 

4430 

4517 

495 

4605 

4693 

4781 

4868 

OO 

4956 

5044 

5131 

5210 

530? 

5394 

496 

5482 

55b9 

5G57 

5744 

5832 

5919 

6007 

1>(W4 

6182 

6269 

497 

6356 

5444 

6531 

6618 

6706 

6793 

6880 

6968 

7055 

7142 

498 

7229 

7317 

7404 

7491 

7578 

7665 

7752 

7839 

7926 

8014 

499 

8101 

8188 

8275 

8362 

8449 

8636 

8622 

8709 

8796 

8883 

OF  NUMBERS.             11 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

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7210 

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688 

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8156 

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1985 

2047  2110 

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4415 

699 

4477 

4539  4601 

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4850 

4912 

4974 

503(5 

OF  NUMBERS.              15 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

700 

845098 

5160 

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5-284 

5346 

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5470 

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7202 

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7819 

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8004 

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62 

705 

8189 

8251 

8312 

8374 

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8743 

706 

8805 

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707 

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9542 

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711 

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2053 

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30-29 

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714 

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16             LOGARITHMS 

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6640 

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6910 

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7026 

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7199 

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1841 

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1955 

2012 

2069 

2126 

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1314 

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1537 

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1705 

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2039 

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6140 

6195 

6251 

6306 

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788 

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6636 

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790 

7627 

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7737 

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8012 

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791 

8176 

8231 

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8396 

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792 

8725 

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9164 

9218 

793 

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9821 

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55 

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0531 

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0640 

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0913 

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1022 

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1131 

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1240 

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1458 

1513 

1567 

1622 

1676 

1736 

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1S94 

1948 

798 

2003 

2057 

2112 

2166 

2221 

2275 

2329 

2384 

2438 

2492 

799 

2547 

2601 

2655 

2710 

2764 

2818 

2873 

2927 

2981 

3036 

OF  NUMBERS.              17 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

800 

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3144 

3199 

3253 

3307 

3361 

3416 

3470 

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801 

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3958 

4012 

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802 

4174 

4229 

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4391 

4445 

4499 

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4661 

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4716 

4770 

4824 

4878 

4932 

4986 

5040 

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6202 

804 

5256 

5310 

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5418 

5472 

5526 

5580 

5634 

5688 

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64 

805 

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5850 

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803 

6335 

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6874 

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7035 

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7143 

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809 

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8056 

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8270 

8324 

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8431 

810 

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8539 

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811 

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812 

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813 

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1424 

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18              LOGARITHMS 

N. 

0 

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856 

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9569 

9616 

9669 

9719 

9769 

9819 

9869 

9918 

9968 

8/1 

940018 

0068 

0118 

0168 

0218 

0267 

0317 

0367 

0417 

0467 

872 

0516 

0566 

0516 

0666 

0/16 

0765 

0815 

0865 

0915 

0964 

873 

1014 

1064 

1114 

1163 

1213 

1263 

1313 

1362 

1412 

1462 

874 

1511 

1561 

1611 

1660 

1710 

1760 

1809 

1859 

1909 

1958 

875 

2008 

2058 

2107 

2157 

2207 

2256 

2300 

2355 

2405 

2455 

876 

2504 

2554 

2603 

26o3 

2702 

2752 

2601 

2851 

2901 

2950 

877 

3000 

3049 

3099 

3148 

3198 

3247 

3297 

3346 

3396 

3445 

878 

3495 

3544 

3593 

3643 

3692 

3742 

3791 

3841 

3890 

3939 

879 

3989 

4038 

4088 

4137 

4186 

4236 

4285 

4335 

4384 

4433 

880 

4483 

4532 

4581 

4631 

4680 

4729 

4779 

4828 

4877 

4927 

881 

4976 

5025 

5074 

5124 

5173 

5222 

5272 

5321 

5370 

5419 

882 

5469 

5518 

5567 

5616 

5665 

5715 

5764 

5813 

6862 

5912 

883 

5961 

6010 

6059 

6108 

6157 

6207 

6256 

6305 

6354 

6403 

884 

6452 

6501 

6551 

6600 

6649 

6698 

6747 

6796 

6845 

6894 

•  885 

6943 

6992 

7041 

7090 

7140 

7189 

7238 

7287 

7336 

7385 

886 

7434 

7483 

7532 

7581 

7630 

7679 

7728 

7777 

7826 

7875  I 

887 

7924 

7973 

8022 

8070 

8119 

8168 

8217 

8266 

8315 

8365 

888 

8413 

8462 

8511 

8560 

8609 

8657 

8706 

8755 

8804 

8853 

889 

8902 

8951 

8999 

9048 

9097 

9146 

9195 

9244 

9292 

9341 

890 

9390 

9439 

9488 

9536 

9585 

9634 

9683 

9731 

9780 

9829 

891 

9878 

9926 

9975 

..24 

..73 

.121 

.170 

.219 

.267 

.316 

892 

950365 

0414 

0462 

0511 

0560 

0308 

0657 

0/06 

0754 

0803 

893 

0851 

0900 

0949 

0997 

1046 

1005 

1143 

1192 

1240 

1289 

894 

1338 

1386 

1435 

1483 

1532 

1580 

1629 

1677 

1726 

1775 

48 

895 

1823 

1872 

1920 

1969 

2017 

2066 

2114 

2163 

2211 

2260 

896 

2308 

2356 

2405 

2453 

2502 

2550 

^599 

2t>4? 

5G96 

2744 

897 

2792 

2841 

2889 

2938 

2986 

3034 

30b3 

3131 

3180 

3228 

898 

3276 

3325 

3373 

34'21 

3470 

3518 

3566 

3bl5 

3663* 

3711 

899 

3760 

3808 

3856 

3905 

3953 

4001 

4049 

4098 

4146 

4194 

OF  NUMBERS.             19 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

900 

954243 

4291 

4339 

4387 

4435 

4484 

4532 

4580 

4628 

4677 

901 

4725 

4773 

4821 

4869 

4918 

4966 

5014 

6062 

6110 

6158 

902 

5207 

5255 

5303 

5351 

5399 

5447 

5495 

5543 

6592 

5640 

903 

5688 

5736 

5784 

6832 

6880 

6928 

5976 

6024 

6072 

6120 

904 

6168 

6216 

6265 

6313 

6361 

6409 

6457 

6505 

6553 

6601 

48 

905 

6649 

6697 

6745 

6793 

6840 

6888 

6936 

6984 

7032 

7080 

906 

7128 

7176 

7224 

7272 

7320 

7368 

7416 

7464 

7512 

7559 

907 

7607 

7655 

7703 

7751 

7799 

7847 

7894 

7942 

7990 

8038 

908 

8086 

8134 

8181 

8229 

8277 

8325 

8373 

8421 

8468 

8516 

909 

8564 

8612 

8659 

8707 

8755 

8803 

8850 

8898 

8946 

8994 

910 

9041 

9089 

9137 

9185 

9232 

9280 

9328 

9375 

9423 

9474 

911 

9518 

9566 

9614 

9661 

9709 

9757 

9804 

9852 

9900 

9947 

912 

9995 

..42 

..90 

.138 

.185 

.233 

.280 

.328 

.376 

.423 

913 

960471 

0518 

0566 

0613 

0661 

0709 

0756 

0804 

0851 

0899 

914 

0946 

0994 

1041 

1089 

1136 

1184 

1231 

1279 

1326 

1374 

915 

1421 

1469 

1516 

1563 

1611 

1658 

1706 

1753 

1801 

1848 

916 

1895 

1943 

1990 

2038 

2085 

2132 

2180 

2227 

2275 

2322 

917 

2369 

2417 

2464 

2511 

2559 

2608 

2653 

2701 

2748 

2795 

918 

2843 

2890 

2937 

2985 

3032 

3079 

3126 

3174 

3221 

3268 

919 

3316 

3363 

3410 

3457 

3504 

3552 

3599 

3646 

3693 

3741 

920 

3788 

3835 

3882 

3929 

3977 

4024 

4071 

4118 

4165 

4212 

921 

4260 

4307 

4354 

4401 

4448 

4495 

4542 

4590 

4637 

4684- 

922 

4731 

4778 

4825 

4872 

4919 

4966 

5013 

5061 

6108 

5155 

923 

5202 

5249 

5296 

5343 

5390 

5437 

5484 

5531 

5578 

5625 

924 

5672 

5719 

5766 

5813 

5860 

5907 

6954 

6001 

6048 

6095 

925 

6142 

6189 

6236 

6283 

6329 

6376 

6423 

6470 

6517 

6564 

926 

6611 

6658 

6705 

6752 

6799 

6845 

6892 

6939 

6986 

7033 

927 

7080 

7127 

7173 

7220 

7267 

7314 

7361 

7408 

7464 

7501 

928 

7548 

7595 

7642 

7688 

7735 

7782 

7829 

7875 

7922 

7969 

929 

8016 

8062 

8109 

8156 

8203 

8249 

8296 

8343 

8390 

8436 

930 

8483 

8530 

8576 

8623 

8670 

8716 

8763 

8810 

8856 

8903 

931 

8950 

8996 

9043 

9090 

9136 

9183 

9229 

9276 

9323 

9369 

932 

9416 

9463 

9509 

9556 

9602 

9649 

9695 

9742 

9789 

9835 

933 

9882 

9928 

9975 

..21 

..68 

.114 

.161 

.207 

.254 

.300 

934 

970347 

0393 

0440 

0486 

0533 

0579 

0626 

0672 

0719 

0765 

935 

0812 

0858 

0904 

0951 

0997 

1044 

1090 

1137 

1183 

1229 

936 

1276 

1322 

1369 

1415 

1461 

1508 

1554 

1601 

1647 

1693 

937 

1740 

1786 

1832 

1879 

1926 

1971 

2018 

20U4 

2110 

2157 

938 

2203 

2249 

2295 

2342 

2388 

2434 

2481 

2527 

2573 

2619 

939 

2666 

2712 

2758 

2804 

2851 

2897 

2943 

2989 

3035 

3082 

940 

3128 

3174 

3220 

3266 

3313 

3359 

3405 

3451 

3497 

3543 

941 

3590 

3636 

3682 

3728 

3774 

3820 

3866 

3913 

3959 

4005 

942 

4051 

4097 

4143 

4189 

4235 

4281 

4327 

4374 

4420 

4466 

943 

4512 

4558 

4604 

4650 

4696 

4742 

4788 

4834 

4880 

4926 

944 

4972 

5018 

6064 

5110 

5166 

5202 

5248 

6294 

5340 

5386 

46 

945 

5432 

5478 

5524 

5570 

5616 

5662 

5707 

5753 

5799 

5845 

946 

5891 

6937 

5983 

6029 

6075 

6121 

6167 

6212 

6268 

6304 

947 

6350 

6396 

6442 

6488 

6533 

6579 

6925 

6671 

6717 

6763 

948 

6808 

6854 

6900 

t3946 

6992 

7037 

7083 

7129 

7175- 

7220 

949 

7266 

7312 

7358 

7403 

7449 

7495 

7541 

7586 

7632 

7678 

=  L  

i 



20              LOGARITHMS 

i 

N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

950 

977724 

7769 

7815 

78G1 

7906 

7952 

7998 

8043 

8089 

8135 

951 

8181 

8226 

8272 

8317 

h363 

8409 

8454 

8500 

8546 

8591 

952 

8637 

8683 

8728 

8774 

8819 

8SG5 

8911 

8956 

9002 

9047 

953 

9093 

9138 

9184 

9230 

9275 

9321 

9366 

9412 

9457 

9503 

954 

9548 

9594 

9639 

9685 

9730 

9776 

9821 

9867 

9912 

9958 

46 

955 

980003 

0049 

0094 

0140 

0185 

0231 

0276 

0322 

0367 

0412 

956 

0458 

0503 

0549 

0594 

0340 

0685 

0730 

0776 

0821 

0867 

957 

0312 

0957 

1003 

1048 

1093 

1139 

1184 

1229 

1276 

1320 

958 

1366 

1411 

1456 

1501 

1547 

1592 

1637 

1683 

1728 

1773 

959 

1819 

1864 

1909 

1964 

2000 

2045 

2090 

2135 

2181 

2226 

960 

2271 

2316 

2362 

2407 

2452 

2497 

2543 

2588 

2633 

2678 

961 

2723 

2769 

2814 

2859 

2904 

2949 

2994 

3040 

3085 

3130 

982 

3175 

3220 

3265 

3310 

3356 

3401 

3446 

3491 

3536 

3581 

963 

3626 

3671 

3716 

3762 

3807 

3852 

3897 

3942 

3987 

4032 

964 

4077 

4122 

4167 

4212 

4257 

43  j2 

4347 

4392 

4437 

4482 

965 

4527 

4572 

4617 

4662 

4707 

4752 

4797 

4842 

4887 

4932 

966 

4977 

5022 

5067 

5112 

5157 

5202 

5247 

6292 

5337 

5382 

967 

6426 

5471 

5516 

5561 

6606 

5651 

5699 

5741 

5786 

5830 

968 

5875 

5920 

5965 

6010 

6055 

6100 

6144 

6189 

6234 

6279 

969 

6324 

6369 

6413 

6458 

6503 

6548 

6593 

6637 

6682 

6727 

970 

6772 

6817 

6861 

6906 

6951 

6996 

7040 

7035 

7130 

7175 

971 

7219 

7264 

7309 

7353 

7398 

7443 

7488 

7532 

7577 

7622 

972 

7666 

7711 

7756 

7800 

7845 

7890 

7934 

7979 

8024 

8068 

973 

8113 

8157 

8202 

8247 

8291 

8336 

8381 

8425 

8470 

8514 

974 

8559 

8604 

8648 

8693 

8737 

8782 

8826 

8871 

8916 

8960 

975 

9005 

9049 

9093 

9138 

9183 

9227 

9272 

9316 

9361 

9405 

976 

9450 

9494 

9539 

9583 

9628 

9672 

9717 

9761 

9806 

9850 

977 

9895 

9939 

9983 

..28 

..72 

.117 

.161 

.206 

.250 

.294 

978 

990339 

0383 

0428 

0472 

0516 

0561 

0605 

0650 

0694 

0738 

979 

0783 

0827 

0871 

0916 

0960 

1004 

1049 

1093 

1137 

1182 

980 

1226 

1270 

1315 

1359 

1403 

1448 

1492 

1536 

1580 

1625 

981 

1669 

1713 

1768 

1802 

1846 

1890 

1935 

1979 

2023 

2067 

982 

2111 

2156 

2200 

2244 

2288 

2333 

2377 

2421 

2465 

2509 

983 

2554 

2598 

2642 

2686 

2730 

2774 

2819 

2863 

2907 

2951 

984 

2995 

3039 

3083 

3127 

3172 

3216 

3260 

3304 

3348 

3392 

985 

3436 

3480 

3524 

3568 

3613 

3657 

3701 

3745 

3789 

3833  : 

986 

3877 

3921 

3965 

4009 

4053 

4097 

4141 

4185 

4229 

4273 

987 

4317 

4361 

4405 

4449 

4493 

4537 

4581 

4625 

4669 

4713 

988 

4757 

4801 

4845 

4886 

4933 

4977 

5021 

5065 

5108 

6152 

989 

5196 

5240 

5284 

6328 

6372 

5416 

6460 

5504 

6547 

6591 

990 

5635 

6679 

6723 

6767 

6811 

5854 

6898 

6942 

5986 

6030 

:  991 

6074 

6117 

6161 

6205 

6249 

6293 

6337 

6380 

6424 

6468 

992 

6512 

6555 

6599 

6643 

6687 

6731 

6774 

6818 

6862 

6906 

993 

6949 

6993 

7037 

7080 

7124 

7168 

7212 

7255 

7299 

7343 

994 

7386 

7430 

7474 

7517 

7561 

7605 

7648 

7692 

7736 

7779 

44 

995 

7823 

7867 

7910 

7954 

7998 

8041 

8085 

8129 

8172 

8216 

996 

8259 

8303 

8347 

8390 

8434 

8477 

8521 

8564 

8608 

8652 

997 

8695 

8739 

8792 

8826 

8869 

8913 

8956 

9000 

9043 

9087 

998 

9131 

9174 

9218 

9261 

9305 

9348 

9392 

9435 

9479 

9522 

999 

9565 

9609 

9652 

9696 

9739 

9783 

9826 

9870 

9913 

9957 

TABLE  II.    Log.  Fines  and  Tangenls.  (0°)  Natural  Sines.          21 

' 

Sine. 

D  10' 

Cosine. 

D.  10" 

Tang. 

D.10' 

Coiang. 

N.sine 

N.  cos. 

0 

0.000030 

10.000000 

0.000000 

Infinite. 

00000 

100000 

60 

6.463726 

oooooo 

6.463726 

13.536274 

00029 

100000 

59 

a 

764753 

oooooo 

764756 

235244 

00058 

103001 

58 

3 

940347 

oooooo 

940847 

059153 

00037 

103000 

57 

4 

7.0ii578o 

oooooo 

7.065786 

12.934214 

001  16 

100000 

56 

5 

162696 

oooooo 

162696 

837304 

00145 

100300 

55 

6 

241877 

9.999999 

241878 

758122 

00175 

100000 

54 

7 

308824 

999999 

308825 

691175 

00204 

100000 

53 

8 

366816 

999999 

366817 

633183 

00233 

100000 

52 

9 

417968 

999999 

417970 

582030 

00262 

100030 

51 

10 

463725 

999998 

463727 

636273 

00291 

100030 

50 

11 

7.505118 

9.999998 

7.505t20 

12.494880 

00320 

99999 

49 

12 

542903 

999997 

542909 

457091 

00349 

9999'J 

48 

13 

577668 

999997 

577672 

422328 

00378 

99999 

47 

14 

609853 

999996 

609857 

390143 

00407 

99999 

46 

15 

639816 

999996 

639820 

360180 

00436 

99999 

45 

16 

667845 

999995 

667849 

332151 

00465 

99999 

44 

17 

694173 

999995 

694179 

305821 

00495 

99999 

43 

1  18 

718997 

999994 

719003 

280997 

00524 

99999 

42 

;  19 

742477 

999993 

742484 

257516 

00553 

99998 

41 

i  20 

764754 

999993 

764761 

235239 

00582 

99998 

40 

21 

7.785943 

9.999992 

7.785951 

12.214049 

00611 

99998 

39 

22 

806146 

999991 

806155 

193845 

00640 

99998 

38 

23 

825451 

999990 

825460 

174540 

00669 

99998 

37 

24 

843934 

999989 

843944 

156056 

00698 

99998 

36 

25 

861663 

999988 

861674 

138326 

00727 

99997 

35 

26 

878695 

999988 

878708 

121292 

00756 

99997 

34 

27 

895085 

999987 

895099 

104901 

00785 

99997 

33 

28 

910879 

999986 

910894 

0391  06 

00814 

99997 

32 

29 

926119 

999985 

926134 

073866 

00844 

99996 

31 

30 

940842 

999983 

940858 

059142 

00873 

99996 

30 

31 

7.955082 

9.999982 

7.955100 

12.044900 

00902 

99996 

•29 

32 

968870 

2298 

999981 

0.2 

968889 

2298 

031111 

00931 

999% 

•28 

33 

982233 

2227 

999980 

0.2 

982253 

2227 

017747 

00960 

99995 

•27 

34 

995198 

2161 

999979 

0.2 

995219 

2161 

004781 

00989 

99995 

•26 

35 

8.007787 

2098 

999977 

0-2 

8.007809 

2098 

11.992191 

01018 

99995 

•25 

36 

020021 

2039 

999976 

0-2 

020045 

2039 

979955 

01047 

99995 

'24 

37 

031919 

1983 

999975 

0-2 

031945 

1983 

968055 

01076 

99994 

'23 

38 

043501 

1930 

999973 

0-2 

043527 

1930 

956473 

01105 

99994 

'22 

39 

054781 

1880 

999972 

0-2 

054809 

1880 

945191 

01134 

99994 

•21 

40 
41 

065776 
8.076500 

1832 

1787 

999971 
9.999969 

0'2 
0-2 

065806 
8.076531 

1833 

1787 

934194 
11.923469 

01164 
01193 

99993 
99993 

'20 
19 

42 
43 

086965 
097183 

1744 
1703 

999968 
999966 

0'2 

o;2 

086997 
097217 

1744 
1703 

913003 
902783 

01222 
01251 

99993 
99992 

18 

17 

44 

107167 

1664 

999964 

0]2 

107202 

1664 

892797 

01280 

99992 

16 

45 

116926 

1626 

999963 

0^3 

116963 

1627 

883037 

01309 

99991 

15 

46 

126471 

1591 

999961 

0^3 

126510 

1591 

873490 

01338 

99991 

14 

47 

135810 

1557 

999959 

0.3 

135851 

1557 

864149 

01367 

99991 

13 

48 
49 
60 
61 
52 
53 
54 
55 
66 
67 

144953 
153907 
162681 
8.171280 
179713 
187985 
196102 
204070 
211895 
219581 

1524 
1492 
1462 
1433 
1405 
1379 
1353 
1328 
1304 
1281 

999958 
999956 
999954 
9.999952 
999950 
999948 
999946 
999944 
999942 
999940 

0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.3 
0.4 

144996 
153952 
162727 
8.171328 
179763 
188036 
196156 
204126 
211953 
219641 

1524 
1493 
1463 
1434 
1406 
1379 
1353 
1328 
1304 
1281 

855004 
846048 
837273 
11.828672 
820237 
811964 
803844 
795874 
788047 
780359 

01396 
01425 
01454 
01483 
01513 
01542 
01571 
01600 
01629 
01658 

99990 
99990 
99989 
99989 
99989 
99988 
99988 
99987 
99987 
99986 

12 
11 
10 
9 
8 
7 
6 
5 
4 
3 

58 

227134 

1259 

999938 

0.4 

227195 

1259 

772805!  01687 

99986 

2 

59 
60 

234557 
241855 

1237 
1216 

999936 
999934 

0.4 
0.4 

234621 
241921 

1238 
1217 

765379  01716 
758079  j  01745 

99985 
99985 

1 
0 

Cosine. 

Sine. 

Coi  an  ST. 

Tang.  H  N.  cos. 

\.  pjn«- 

' 

89  Degrees. 

22          Log.  Sines  and  Tangents.  (1°)  Natural  Sines.   TABLE  II. 

' 

Sine. 

D.10" 

Cosine. 

D.10" 

Tan£ 

D  10" 

Coiang.  i'N.  sine. 

N.  cos. 

o 

8.241855 

9.999934 

8.241921 

11.758079 

01742 

J9985 

60 

1 

249033 

1196 

999932 

0.4 

249102 

197 

750898 

01774 

)9984 

59 

2 

256094 

117/ 

999929 

0-4 

256165 

.177 

1  1  KQ 

743835 

01803 

J99S4 

58 

3 

263042 

1158 

999927 

0.4 

0,4 

263115 

L15o 
1  1  1  n 

736885 

01832 

99983 

57 

4 

269881 

1140 

999925 

.4 

269956 

L14U 

730044 

01862 

99983 

56 

5 

276614 

1122 

999922 

0.4 

276691 

L122 

723309 

01891 

)9982 

55 

6 

283243 

1105 

999920 

0.4 

283323 

1105 

716677 

01920 

99982 

54 

7 

289773 

1  083 

999918 

0.4 

289856 

1089 

710144 

01949 

9<>y8i 

53 

8 

29620  7 

10/2 

999915 

0.4 

OA 

296292 

1073 

1  AX? 

703708 

01978 

99980 

52 

9 
10 

302541) 
308794 

1056 
1041 

999913 
999910 

.4 

0-4 

302634 

308884 

lUo/ 
1042 

1  10  1 

697366 
691116 

02007 
02036 

99980 
99979 

51 
50 

11 

8.314954 

1027 

9.999907 

0  -4 

04 

8.315046 

lU4/ 

1  01  ^ 

11-684954 

02065 

9B979 

49 

12 

321027 

OAQ 

999905 

.  4 

04 

321122 

lUlo 

AGO 

678878 

02094 

99978 

48 

13 

327016 

yyo 

no^ 

999902 

•  4 

OA 

327114 

yyt-j 

672886 

02123 

99977 

47 

14 

332924 

yoo 

Q71 

999899 

.4 

OK 

333025 

985 
070 

666975 

02152 

9U977 

46 

15 

338753 

y  /  1 

QXQ 

999897 

.  o 

OK 

333856 

J  t  M 
9XQ 

661144 

02181 

9997(5 

46 

16 

344504 

yoy 

999894 

.  O 
OK 

344610 

OJ 

655390 

02211 

99976 

44 

17 

350181 

Q^A 

999891 

.  O 
OK 

350289 

„„  , 

649711 

02240 

99975 

•l:> 

18 

355783 

Q99 

999888 

•  O 
OK 

355895 

922 

644105 

02269 

99974 

42 

19 

361315 

Q1O 

999885 

•O 

OK 

361430 

qi  i 

638570 

02298 

99974 

41 

20 

366777 

QQQ 

999882 

.O 
OK 

366895 

yi  i 

8QQ 

633105 

02327 

99973 

40 

21 

8.372171 

oyy 

ooo 

9.999879 

.  O 

OK 

8.372292 

QJiJ 

888 

11-627708 

02356 

9y972 

39 

22 

377499 

OOO 
877 

999876 

.O 

OK 

377622 

OOO 

87Q 

622378 

02385 

99972 

38 

23 

382762 

Oil 

999873 

•O 
OK 

382889 

/y 

617111 

02414 

99971 

37 

24 

387962 

8"R 

999870 

,O 

OK 

388092 

857 

611908 

02443 

99970 

36 

25 

393101 

84fi 

999867 

.  U 
A  K 

393234 

847 

606766 

02472 

99969 

35 

26 

398179 

007 

999864 

V  •  O 
OK 

398315 

837 

601685 

02501 

99969 

34 

27 

403199 

Oo  / 

897 

999861 

.  O 
OK 

403338 

828 

596662 

02530 

99968 

33 

28 

408161 

oz  / 
818 

999858 

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OK 

408304 

818 

691696 

02560 

99967 

32 

29 

413068 

Olo 

999854 

.  O 
OK 

413213 

QAQ 

686787 

02589 

99966 

31 

30 

417919 

800 

999851 

.  O 
Of; 

418068 

ouy 

800 

681932 

02618 

99966 

30 

31 

8.422717 

7O.1 

9.999848 

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8.422869 

ouu 

11.577131 

02647 

99965 

29 

32 

427462 

/yi 

782 

999844 

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0  6 

427618 

783 

672382 

02676 

99964 

28 

33 

432156 

/  o/* 

774 

999841 

u  .  o 

Of; 

432315 

567685 

02705 

99963 

27 

34 

436800 

/  /4 
7fjf> 

999838 

•  O 
Of; 

436962 

7KK 

563038 

02734 

99963 

26 

35 

441394 

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758 

999834 

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OR 

441560 

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758 

558440 

02763 

99962 

25 

36 

445941 

/oo 

7nO 

999831 

.  O 
OR 

446110 

750 

553890 

02792 

99961 

24 

37 

450440 

/OU 

999827 

.  O 

Of! 

450613 

649387 

02821 

99960 

23 

38 

454893 

70K 

999823 

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OR 

455070 

735 

544930 

02850 

99959 

22 

39 

459301 

/GO 

999820 

•  O 

Or> 

459481 

ryno 

640519 

02879 

99959 

21 

40 

463666 

7on 

999816 

.0 

Of; 

463849 

/  ^Q 

720 

636151 

02908 

99958 

20 

41 

8.467985 

719 

9.999812 

•  O 
Of; 

8.468172 

/^u 
71  ^5 

11.531828 

02938 

99957 

19 

42 

472263 

/  1-4 

999809 

•O 
O.fi 

472454 

/  lo 
707 

627546 

02967 

99956 

18 

43 

476498  '™ 

999805 

O 
Of; 

476693 

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700 

623307 

02996 

99955 

17 

44 

480693  ;££ 

999801 

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Of; 

480892 

/  uu 

619108 

03025 

99954 

16 

45 

484848  i  5 

999797 

.0 

485050 

.  >  , 

614950 

03054 

99953 

15 

46 

4889631  ;£n 

999793 

0-7 

Ort 

489170 

fifiO 

610830 

03083 

99952 

14 

47 

4930401  S«q 

999790 

•  1 

493250 

OoU 

506750 

03112 

99952 

13 

48 

497078  SA, 

999786 

0.7 

Ort 

497293 

fi  •« 

602707 

03141 

99951 

12 

49 

501080  ggj 

999782 

•  / 
Ory 

601298 

fifi! 

498702 

03170 

99950 

11 

50 

605045  r  _  „ 

999778 

.  / 

07 

605267 

OO1 

655 

494733 

03199 

99949 

10 

51 

8.508974  5 

9.99977..' 

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0*7 

8.509200 

£?KA 

11.490800 

03228 

99948 

9 

52 

512867  i  «™ 

999769 

.  / 

017 

613098 

OOU 

486902 

03257 

99947 

8 

53 

616726  J™ 

999765 

.  / 

017 

516961 

f?00 

483039 

03286 

99946 

7 

54 

520551  gg 

999761 

.  / 

07 

520790 

OoO 

fjOQ 

479210 

03316 

99945 

6 

55 

524343  %£ 

999757 

.  / 

On 

524586 

OOO 

CiOT 

475414 

03345 

99944 

6 

66 

528102  %£ 

999753 

.7 

Ort 

528349 

O^i  / 

471651 

03374 

99943 

4 

67 

531828  jjf* 

999748 

.  / 
Orr 

532080 

filfi 

467920 

03403 

99942 

3 

58 

635523  p}. 

999744 

.  1 
0*7 

635779 

010 
f»1  1 

464221 

03432 

99941 

2 

59 

639186  °,* 

999740 

.  / 

07 

539447 

Ol  1 

fiOfi 

460553 

03461 

99940 

1 

60 

542819  C 

999735 

.  / 

543084 

ouo 

466916 

03490 

99939 

0 

Cosine. 

Sine. 

Cot  a  n  «;. 

Tang. 

N.  cos. 

N.sine. 

' 

88  Degrees. 

TABLE  II.    Log.  Sines  and  Tangents.  (2°)  Natural  Sines.           23 

' 

Sjne. 

D.  10" 

Cosine. 

D.  10' 

Tang.  |D.  10' 

Cotang.  IN.  sinc.|N.  cos 

GO 

0 

8.542819 

9.999735 

07 

8.543084 

11.456916  i  03490 

99939 

1 

546422 

bUU 

999731 

.  / 

546691 

DUZ 

4533091|03519 

99938 

59 

.2 

549995 

595 

999726 

0.7 

550268 

593 

449732!  103548 

9993/ 

58 

3 

553539 

591 

9997-22 

0-7 

553817 

591 

446183!  03577 

9')93i 

57 

4 

557054 

!?  ) 

999717 

0-8 

557336 

or>7 

442664  j  03606 

99935 

56 

5 

560540 

581 

999713 

0-8 

600828 

582 

439172 

03635 

99934 

55 

6 

663999 

576 

999708 

0-8 

564291 

577 

435709 

03664 

99933 

54 

7 

667431 

572 

999704 

0.8 

507727 

573 

432273 

03693 

99932 

53 

8 

670836 

507 

999099 

0-8 

00 

671137 

568 
KP  1 

428863 

03723 

99931 

52 

9 

574214 

KKQ 

999694 

•  b 

00 

574520 

Oo4 

K.KC1 

425480 

03752 

99931 

51 

10 

577506 

KZ.A 

999689 

•O 

OQ 

577877 

ooy 

K*-  X 

422123 

03781 

99929 

5C 

11 

8.580892 

OO4 

KKA 

9.999685 

•  O 

00 

8.581208 

Ooo 

K  -  1 

11.418792 

03810:99927 

49" 

12 

684193 

OOU 

999080 

•O 
00 

584514 

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415486 

03839 

99920 

48 

13 

687469 

546 

999675 

•O 

687795 

647 

412205  ! 

03808 

99925 

47 

14 

690721 

542 

COO 

999070 

0.8 

0O 

591051 

643 

408949!!  03897 

99924 

46 

15 

693948 

Ooo 

999665 

.8 
0O 

694283 

539 

405717 

03926 

99923 

45 

16 

597152 

OJ4 

9996(50 

.8 

697492 

535 

402508 

03955 

99922 

44 

17 

600332 

530 

999055 

0.8 

600077 

631 

399323 

03984 

99921 

43 

18 

603489 

526 

999650 

0.8 

00 

603839 

627 

396161 

04013 

99919 

42 

19 

606623 

e  -i  (\ 

999645 

.8 

Of. 

608978 

523 

393022 

04042 

99918 

41 

20 

609734 

oiy 
~i  r* 

999640 

.8 

0.-. 

610094 

519 

389906 

04071 

99917 

40 

21 

8.612823 

OlO 

C1  1 

9.999635 

•  9 

On 

8.613189 

616 

11.386811 

04100 

99916 

39 

22 

615891 

Ol  1 

KAQ 

999629 

•9 
On 

616262 

K  « 

383738 

03129 

99915 

38 

23 

618937 

OUO 

999324 

•  9 

619313 

508 

380687 

04159 

99913 

37 

24 

621962 

KA1 

999619 

0-9 

622343 

505 

377657 

04188 

99912 

36 

25 

624965 

OU1 

999614 

0-9 
On 

625352 

601 

374648 

04217 

99911 

35 

26 

627948 

4ol 

999608 

•9 

On 

628340 

498 

371660 

04246 

99910 

34 

27 
28 

630911 
633854 

490 

AW7 

999603 
999597 

•9 

0-9 
On 

631308 
634256 

496 
491 

368692 
365744 

04275  99909 
04304(99907 

33 
32 

29 

636776 

4O  / 

,40  A 

999592 

•  9 

637184 

488 

362816 

04333|99900 

31 

30 

639680 

4o4 

A  W1 

999586 

0.9 

On 

640093 

485 

369907 

04362)99905 

30 

31 

8.642563 

4ol 

AT7 

9.999581 

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On 

8.642982 

482 

11.357018 

04391(99904 

29 

32 
33 

645428 

648274 

4  /  / 

474 
471 

999575 
999570 

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0.9 

0  9 

645853 
648704 

478 
475 

354147 
351296 

0442099902 
04449  99901 

28 

27 

34 

.651102 

T:  /  1 

999564 

On 

651537 

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348463 

04478 

99900 

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35 
30 

653911 
656702 

465 
4K2 

-  999558 
999553 

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•  0 

654352 
657149 

409 
466 

345648 
342851  1 

04507 
04536 

99898 
99897 

25 

24 

37 

659475 

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999547 

659928 

463 

340072 

04565- 

99896 

23 

38 

602230 

A~r 

999541 

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662089 

460 

A  -o1 

337311 

04594 

J9894 

22 

39 

664908 

d~i 

999535 

665433 

4o7 

A  r  A 

334567 

04623 

99893 

21 

40 

667689 

4">i 

999529 

In 

668160 

454 

331840  i 

04653 

99892 

20 

41 
42 

8.670393 
673080 

448 

A  A^ 

9.999524 
999518 

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1-0 

1(\ 

8.670870 
673563 

453 
449 

11.  329130  i 
326437  1 

04682 
04711 

99890 
99889 

19 
18 

43 

675751 

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A  AC) 

999512 

•U 

676239 

446 

323761 

04740 

99888 

17 

44 
45 

678405 
681043 

44,* 

440 

999506 
999500 

1  -0 
1-0 

678900 
681544 

443 

442 

321100 
318456 

04769 
04796 

99886 
99885 

16 
15 

46 

683665 

4d7 

999493 

1  .0 

684172 

438 

315828  !l  04827 

99883 

14 

47 

686272 

434 

999487 

1.0 

6  6784 

435 

313216,104850 

99882 

13 

48 

688863 

4OQ 

999481 

1  .0 

1A 

689381 

433 

310619 

04885 

99881 

12 

49 

691438 

497 

999475 

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In 

691963 

430 

308037 

04914 

99879 

11 

60 

693998 

*i^>  1 

999469 

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1ft 

694529 

428 

305471  !  04943 

99878 

10 

51 

8.696543 

499 

9.999463 

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1-1 

8.697081 

425 

11.302919  04972 

99876 

9 

62 

699073 

%£• 

999456 

.  1 

699617 

423 

300383 

05001 

99875 

8 

63 

701589 

417 

999450 

1  .1 

702139 

420 

297861 

05030 

99873 

7 

64 

704090 

414 

999443 

1  . 

704246 

418 

295354 

05059 

99872 

6 

65 

706577 

419 

999437 

1  . 

707140 

415 

292860  1 

05088 

99870 

5 

66 

709049 

A  1  A 

999431 

1  . 

709618 

413 

290382  05117 

99869 

4 

67 

711507 

4.1  U 

999424 

1  . 

702083 

411 

2879171  05146 

99807 

3 

68 
59 

713952 
716383 

405 

999418 
999411 

1 
1. 

714534 
716972 

408 
406 

285465  i  05175 
283028  05205 

99866 
99864 

2 

1 

60 

718800 

403 

999404 

1  .1 

719396 

404 

280604  05234 

99863 

0 

Cosine. 

SnTe! 

Cotang. 

Tang.   Jj  N.  cos. 

N.sine 

~1~ 

87  Degrees. 

Log.  Sines  and  Tangents.  (3°;  Natural  Sines.   TABLE  II. 

' 

Sine. 

D.  lu' 

Cosine. 

D.  10' 

Tang. 

D.  10' 

Cotang.  |(N.  sine 

N.  cos 

0 

8.718800 

.101 

9.999404 

8.719396 

4(V> 

11.280604 

05234 

99863 

60 

1 

721204 

4U1 

OQQ 

999398 

1" 

721806 

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oqq 

278194 

05263 

99861 

59 

2 

723595 

oyo 

999391 

724204 

oyy 
00-7 

275796 

05292 

99860 

58 

3 

725972 

396 

QO  -i 

999384 

.1 

726588 

d97 

OQK 

273412 

05321 

99858 

57 

4 

728337 

oy^i 

QQO 

999378 

728959 

oyo 

OQO 

271041 

05350 

99857 

56 

5 

730688 

o>y^ 

999371 

731317 

oyo 

268683 

05379 

99855 

55 

6 

733027 

390 

OQO 

999364 

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733663 

391 

OQQ 

266337 

05408 

99854 

54 

7 

735354 

ooo 

999357 

735996 

ooy 

264004 

05437 

99852 

53 

8 

737667 

386 

999350 

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738317 

387 

261683 

05466 

99851 

52 

9 

739969 

384 

999343 

.2 

740326 

385 

259374 

05495 

9984S 

51 

10 

742259 

382 

999336 

.  2 

742922 

383 

OO  1 

257078 

05524 

99847 

50 

11 

8.744536 

380 

9.999329 

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8.745207 

dol 

11.254793 

05553 

9984G 

49 

12 

746802 

378 

999322 

.2 

747479 

379 

252521 

05582 

99844 

48 

13 

749055 

376 

999315 

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749740 

377 

250260 

05611 

99842 

47 

14 

751297 

374 

999308 

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751989 

375 

248011 

05640 

99841 

46 

15 

753528 

372 

999301 

.2 

754227 

373 

245773 

05669 

99839 

45 

16 

755747 

370 

999294 

.2 

756453 

371 

243547 

05698 

99838 

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17 

757955 

368 

999286 

.2 

758668 

369 

241332 

05727 

99836 

43 

18 

760151 

366 

999279 

.2 

760872 

367 

239128 

05756 

99834 

43 

19 

762337 

364 

999272 

.2 

763065 

365 

236935 

05785 

99833 

41 

20 

764511 

362 

Q£M 

999265 

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765246 

364 

or*O 

234754 

05814 

99831 

40 

21 

8.766675 

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9.999257 

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11.232583 

05844 

99829 

39 

22 

768828 

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999250 

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230422 

05873 

99827 

38 

23 

770970 

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999242 

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228273 

05902 

99826 

37 

24 

773101 

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773866 

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226134 

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99824 

36 

25 

775223 

353 

O"O 

999227 

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224005 

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99822 

35 

26 

777333 

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999220 

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778114 

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221886 

05989 

99821 

34 

27 

779434 

350 

999212 

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219778 

06018 

99819 

33 

28 

781524 

348 

999205 

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782320 

350 

O  AQ 

217680 

06047 

99817 

32 

29 

783605 

347 

999197 

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215592 

06076 

99815 

31 

30 

785675 

345 

999189 

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786486 

346 

213514 

06105 

99813 

30 

31 

8.787736 

343 

9.999181 

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8.788554 

345 

11.211446 

06134 

99812 

29 

32 

789787 

342 

999174 

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343 

209387 

06163 

99810 

28 

33 

791828 

340 

999166 

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792662 

341 

207338 

06192 

99808 

27 

34 

793859 

339 

999158 

.3 

794701 

340 

205299 

06221 

99806 

26 

35 

795881 

337 

999150 

.3 

796731 

338 

OO'T 

203269 

06250 

99804 

25 

36 

797894 

335 

999142 

•  3 

798752 

OO  / 
OOK 

201248 

06279 

99803 

24 

37 

799897 

334 

999134 

•  3 

800763 

ooo 

199237 

06308 

99801 

23 

38 

801892 

332 

999126 

•  3 

802765 

334 

197235 

06337 

99799 

22 

39 

803876 

331 

999118 

•  3 

804858 

332 

195242 

06366 

99797 

21 

40 

805852 

329 

999110 

.3 

806742 

331 

193258 

06395 

99795 

20 

41 

8.807819 

328 

9.999102 

•  3 

8.808717 

329 

11.191283 

06424 

99793 

19 

42 

809777 

326 

999094 

•  3 

810683 

328 

189317 

06453 

99792 

18 

43 

811726 

325 

999086 

1.4 

812641 

326 

187359 

06482 

99790 

17 

44 

813667 

323 

999077 

1.4 

814589 

325 

185411 

06511 

99788 

16 

45 

815599 

322 

oon 

999069 

1.4 

816529 

323 

OOO 

183471 

06540 

99786 

15 

46 

817522 

o20 
010 

999061 

1.4 

818461 

622 

181539 

06569 

99784 

14 

47 

819436 

o!9 

01  o 

999053 

1.4 

820384 

320 

179616 

06598 

99782 

13 

48 

821343 

olo 
01  c° 

999044 

1*4 

822298 

319 

177702 

06627 

99780 

12 

49 

823240 

olb 

O1  K 

999036 

1  .4 

824205 

318 

176795 

06656 

99778 

11 

60 

825130 

olO 

999027 

1.4 

826103 

316 

173897 

06685 

99776 

10 

51 

8.827011 

313 

9.999019 

1.4 

8.827992 

315 

11.172008 

06714 

99774 

9 

52 

828884 

312 

999010 

1.4 

829874 

314 

170126 

06743 

99772 

8 

53 

830749 

311 

999002 

1.4 

831748 

312 

168252 

06773 

99770 

7 

54 

832607 

309 

998993 

1.4 

833613 

311 

166387 

06802 

99768 

6 

55 

834456 

308 

998984 

1.4 

835471 

310 

164529 

06831 

99766 

5 

56 

836297 

307 

998976 

1.4 

837321 

308 

162679 

06860 

99764 

4 

57 

838130 

306 

998967 

1.4 

839163 

307 

160837 

06889 

99762 

3 

58 

839956 

304 

998958 

1.5 

840998 

306 

159002 

06918 

99760 

2 

59 

841774 

303 

998950 

1.5 

842826 

304 

157175 

06947 

99758 

1 

60 

843585 

302 

998941 

1.5 

844644 

303 

155356 

06976 

99766 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

N.sine. 

i 

86  Degrees. 

TABU-:  II.    Log.  Sines  and  Tangents.  (4°)  NaUral  Sines.          25 

'  > 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang. 

N.  sine. 

N.  cos. 

0 

8.843585 

OAA 

9.998941 

1  K 

8.844644 

onr> 

11.155356 

06976 

99756 

60 

1 

845387 

OUU 
OOQ 

998932 

1  .  0 

Ic; 

846455 

oUw 

OA1 

153545 

07005 

99754 

59 

2 

847183 

Zyy 

OQO 

998923 

.  O 

IK 

848260 

OU1 
9QQ 

151740 

07034 

99752 

58 

3 

848971 

zyo 

9Q7 

998914 

•  O 

IK 

850^57 

zyy 

9Q8 

149943 

07063 

99750 

57 

4 

850751 

Zy  I 
9QK 

998905 

•O 

IK 

851846 

zyo 

9Q7 

148154 

07092 

99748 

56 

5 

852525 

zyo 

9QJ. 

998896 

•0 

IK 

853628 

*  •  '  » 

9Q:' 

146372 

07121 

99746 

55 

6 

854291 

zy*± 

OQQ 

998887 

•  0 

IK 

855403 

zy.  > 

OGK. 

144597 

07150 

99744 

54 

7 

856049 

zyo 

OOO 

998878 

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IK, 

857171 

zyo 

OO(? 

142829 

07179 

99742 

53 

8 

857801 

ZyZ 

291 

998869 

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1  K 

858932 

zyj 

292 

141068 

07208 

99740 

52 

9 

859546 

9;)0 

998860 

1  -  O 
1  fi 

860686 

901 

139314 

07237 

99738 

51 

10 

861283 

Z'.>u 

o«« 

998851 

i  •  O 
1  fi 

862433 

ZJl 

O(i() 

137567 

07266 

99736 

50 

11 

8.863014 

zoo 
os  7 

9.998841 

1  •  O 
1  fi 

8.864173 

<*<j\j 
oco 

11.135827 

07295 

99734 

49 

12 

864738 

ZO  i 

286 

998832 

1  •  0 

IK 

865906 

^oy 

000 

134094 

07324 

99731 

48 

13 

866455 

zoo 
oes 

998823 

•  D 
1  R 

867632 

Zoo 
987 

132368 

07353 

99729 

47 

14 

868165 

zoo 
98.4. 

998813 

JL  .O 

.:? 

869351 

Zo  / 

9ox 

130649 

07382 

99727 

46 

15 

869868 

ZO4 
9«Q 

998804 

,o 

871064 

ZoO 

98  d 

128936 

07411 

99725 

45 

16 

871565 

Zoo 
989 

998795 

872770 

ZO4 
981 

127230 

07440 

99723 

44 

17 

873255 

ZOZ 

981 

998785 

874469 

Zoo 
989 

125531 

07469 

99721 

43 

18 

874938 

ZoJ. 
279 

998776 

_ 

g 

876162 

Zoz 
981 

123838 

07498 

99719 

42 

19 

876615 

279 

998766 

877849 

A^Ol 

980 

122151 

07527 

99716 

41 

20 

878285 

277 

998757 

f; 

879529 

^•OU 
070 

120471 

07556 

99714 

40 

21 

8.879949 

276 

9.998747 

•  U 

8.881202 

z  i  y 
978 

11.118798 

07585 

99712 

39 

22 

881607 

97'"i 

998738 

882869 

Z  /O 

9,77 

117131 

07614 

59710 

38 

23 

883258 

«•<•«) 

974 

998728 

884530 

Z  1  / 

97fi 

115470 

07643 

99708 

37 

24 

•884903 

Z  /^* 
271 

998718 

886185 

z  ^o 
971; 

113815 

07672 

99705 

36 

25 

886542 

/o 

979 

998708 

887833 

Z  4  O 
974 

112167 

07701 

99703 

35 

26 

888174 

4tA 

971 

998699 

c 

889476 

Z  <  4 
97Q 

110524 

07730 

99701 

34 

27 

889801 

A  /  1 

270 

998689 

•  D 

•6 

891112 

Z  io 

272 

108888 

07759 

99699 

33 

28 

891421 

9fiO 

998679 

892742 

z  iz 
971 

107258 

07788 

)9696 

32 

29 

893035 

ZoJ 

268 

998669 

7 

894300 

Z/  1 

970 

105634 

07817 

99694 

31 

30 

894643 

Of;7 

998659 

895984 

*  1  V 

9AQ 

104016 

07846 

99(592 

30 

31 

8.896246 

zo  i 

°(U5 

9.998649 

7 

8.897596 

zoy 

OfiQ 

11.102404 

07875 

99689 

29 

32 

897842 

wOl) 

9  /jr. 

998639 

7 

899203 

^-oo 
2fi7 

100797 

07904 

J9687 

28 

33 

899432 

ZOO 
O*^  <1 

998629 

.  / 

r* 

900803 

ZD  1 

Of?£; 

099197 

07933 

#686 

2~ 

34 

901017 

ZO4 
t>i'*j 

998619 

.  / 

902398 

ZOO 

O^'  »X 

097602 

079(52 

'9683 

26 

35 

902596 

iioo 

O/  'O 

998609 

903987 

ZOO 
OdA 

096013 

07991 

)9680 

25 

3i3 
37 

904169 
905736 

ZOZ 

261 

Ofjf) 

998599 
998589 

.7 

•  7 
•  7 

905570 
907147 

Zo4 
263 
262 

094430 
092853 

08020 
08049 

99678]  24 

99676  23 

38 

907297 

.  ZOU 
OXQ 

998578 

908719 

9fi1 

091281 

08078 

99673  22 

39 

908853 

^Oa 

9^8 

998568 

7 

910285 

zol 
260 

089715 

08107 

99671  21 

40 
41 

910404 
8.911949 

zoo 
257 

9n7 

998558 
9.998548 

•  / 

.7 

7 

911846 
8.913401 

zou 
259 

OKO 

088154 
11.086599 

08136 
08165 

996(58 
9966(5 

20 
19 

42 

913488 

~  •  )  / 

9^fi 

998537 

.  / 

7~ 

914951 

ZOo 
0x7 

085049 

08194 

99664 

18 

43 

915022 

*oo 

OKK 

998527 

916495 

zo  / 

9!Sfi 

083505 

08223 

99661 

17 

44 

916550 

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OK/4 

998516 

918034 

ZOO 

OKC 

081966 

08252 

99(559 

16 

45 

918073 

^04 

2,%o 

998506 

919568 

zoo 

9n^i 

080432 

08281 

99657 

15 

46 

919591 

Do 

998495 

_ 

921096 

ZOO 

O"  A 

078904 

08310 

99(554 

14 

47 

921103 

252 

998485 

922619 

zo4 

077381  !  08339 

99652 

13 

48 

922(510 

251 

998474 

.8 

9241  36 

253 

2-^0 

075864 

08368 

99649 

12 

49 

9-J4112 

250 

998464 

.8 

925649 

02 

074351 

08397 

99647 

11 

5.') 

925(509 

249 

o  in 

998453 

.8 
g 

927156 

251 
OPCA 

072844 

08426 

99644 

10 

51  8.927100 

f£'-±iJ 

9.998442 

8.928658 

ZJU 

11.071342 

08455 

99642 

9 

928587 

248 

o/i  re 

998431 

.8 

930155 

249 

O  in 

069845 

08484 

99639 

8 

53 

930068 

!*47 
O/ift 

998421 

0 

931647 

^4y 

O/lrt 

068353 

08513 

99637 

7 

51 

931544 

^i*iO 
^AK 

998410 

•  O 

933134 

Z^lo 
917 

066866 

08542 

99635 

6 

55 

!)33015 

,w4:O 

.  )  1  1 

998399 

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9346  1(5 

Z-l  / 
"  1C 

065384  08571 

99632 

5 

56 

934481 

Z'\~± 

Q  1Q 

T98388 

_ 
y 

936093 

«-  M  > 

245 

063907 

(IS(iOl) 

99630 

4 

57 

:  ;:;r  ..:-. 

^UO 

998377 

937565 

O  i  1 

062435 

08629 

99627 

3 

58 

937398 

fcJ43 

o  10 

998366 

.8 

939032 

z44 
9J.J. 

060968 

08658 

99625 

2 

59 
60 

938S50 
94u296 

^••4^ 

241 

998355 
998344 

'.b 

940494 
941952 

Z44 

243 

059506 
058048 

08687 
08716 

99622 
99619 

1 
0 

(losino. 

SJneT~ 

Co  tang. 

Tang.   il  N.  cos. 

N.sinc. 

/ 

85  Degrees. 

26           Log.  Sines  and  Tangents.  (5°)  Natural  Sines.    TABLE  11. 

' 

Sine.   D.  10"  Cos-ine. 

D.  10' 

Tang. 

D.  10 

Cotang.  |  jN.  sine 

X.  COR 

0 

8.940296 

9.998344 

8.941952 

11.058048  08716 

9961 

60 

1 

941738 

Z4U 

993333 

1  .9 

943404 

242 

056596 

08745 

9961 

59 

2 

943174 

239 

998322 

1  .9 

944852 

241 

055148 

08774 

99614 

58 

3 

944606 

« 

998311 

1.9 

946295 

240 

053705 

08803 

99612 

57 

4 

946034 

238 

998300 

1  .9 

947734 

240 

052266 

08831 

99609 

56 

5 

947456 

26  i 

998289 

1  .9 

949168 

239 

050832 

08860 

9960* 

55 

6 

948874 

236 

998277 

1.9 

950597 

238 

049403 

08889 

99804 

54 

7 

950287 

00- 

998266 

1  .9 

1  Q 

952021 

937 

047979 

08918 

99602 

53 

8 

951698 

234 

998255 

i  .  y 
1q 

953441 

Zo  / 

046559 

08947 

99599 

52 

9 

953100 

933 

998243 

•  y 

1  Q 

954856 

0__ 

045144 

08976 

9959( 

51 

10 

954499 

Zoo 

998232 

i  .  y 
1  q 

956267 

934 

043733 

09005 

99594 

50 

11 

8.955894 

9.39 

9.998220 

i  ,  y 
1q 

8,957674 

93d 

11.042326 

09034 

99591 

49 

12 

957284 

. 

998209 

.  y 

1  Q 

959075 

QOO 

040925 

09063 

99588 

48 

13 

958670 

930 

998197 

i  .  y 

1  Q 

960473 

Zoo 

039527 

09092 

99586 

47 

14 

960052 

ZoU 

998186 

i  .  y 

1r\ 

961866 

232 

038134 

109121 

99583 

46 

15 

961429 

OQ 

998174 

.9 

If-* 

963255 

~31 

036745 

09150 

99580 

45 

16 

962801 

228 

998163 

.9 
t  q 

964639 

231 

035361 

09179 

99578 

44 

17 

964170 

227 

998151 

A  *  y 

1  Q 

966019 

92Q 

033981 

09208 

99575 

43 

18 

965534 

997 

998139 

JL  ,  y 

0  A 

967394 

99Q 

032606 

09237 

99572 

42 

19 

966893 

ZZ  i 

998128 

Z  .  \) 

2  0 

968766 

9OQ 

031234 

09266 

99570 

41 

!  20 

968249 

99" 

998116 

2  A 

970133 

997 

029867  |09295 

99567 

40 

21 

8.969600 

994. 

9.998104 

.  u 

O  A 

8.971496 

ZZ  / 

11.028504 

09324 

99564 

39 

22 

970947 

ZZ4 

998092 

9  ft 

972855 

99fi 

027145 

09353 

)95(>2 

38 

23 

972289 

223 

998080 

Z.  U 
2  0 

974209 

90  K 

025791 

09382|99559 

37 

24 

973628 

998068 

z  .  u 
2/1 

975560 

zzo 

024440 

09411199556 

36 

25 

974962 

999 

998056 

.  U 

O  A 

976906 

91 

023094 

0944099553 

35 

26 

976293 

99  \ 

998044 

z  ,  \) 

2  A 

978248 

993 

021752 

09469(99561 

34 

27 

28 

977619 
978941 

220 

998032 
998020 

•  V 

2.0 

2   A 

979586 
980921 

222 

0204141  0949899548 
019079  09527199545 

33 
32 

29 

980259 

91  Q 

998008 

.  U 

9  O 

982251 

901 

0  1  7749  I  09556 

99542 

31 

30 
31 

32 

981573 

8.982883 
984189 

ziy 
218 
218 
217 

997996 
9.997984 
997972 

Z.  U 

2.0 

2.0 
2  0 

983577 
8.984S99 
986217 

220 
220 

016423!  09585 
11.015101  J09614 
013783:109642 

99540 
99537 
99534 

30 
29 

28 

I  33 

985491 

**i  i 

997959 

o'n 

987532 

f)  1  U 

012468J  09671)99531 

27 

34 

988789 

Ol  Q 

997947 

^  .  U 
O  t\ 

988842 

011158  09700(99528 

26 

35 

988083 

O1  ^ 

997935 

O  1 

990149 

01  7 

009851 

09729  99526 

25 

36 
37 

989374 
990660 

*>  10 
214 
01  \ 

997922 
997910 

•*  •  i 

2.1 

21 

991451 
992750 

-  !  i 

216 

008549 
007250 

06758  99523 
0978799520 

24 
23 

38 

991943 

z!4 
21  3 

997897 

.1 
2  1 

994045 

216 
91  K 

005955  |09816J99517 

22 

39 

993222 

Z1O 

91  o 

997885 

Z.I 

0  i 

995337 

Z1O 

004663 

09845  99514 

21 

40 

994497 

Zl^ 
OlO 

997872 

Z.I 
9  1 

996624 

O1  A 

003376 

0987499511 

20 

41 
42 

43 
44 
45 

S.  995  768 
997036 
998299 
999560 
).  0008  16 

o  to  to  to  to  t 

E  co  o  i—  '  i—  >  t 

J.  997860 

997847 
997835 
997822 
997809 

o  to  to  to  to  t 

8.997908 
999188 
9.000465 
001738 
003007 

o  to  to  to  to  t 

-  I—  tO  CO  CO  ,4 

11.002092 
000812 
0.999535 
998262 
996993 

0990399508 
09932199508 
09961  99503 
09990,99500 
10019(99497 

19 
18 
17 
16 
15 

46 

002069 

0 

997797 

24 

004272 

**!.*. 

995728 

10048199494 

14 

47 

003318 

9/vo 

997784 

.1 
21 

005534 

210 

994466 

10077|99491 

13 

48 

004563 

ZUo 
9/17 

997771 

.  1 

O  1 

006792 

0  Q 

993208  1 

10108i99488 

12 

49 

005805 

ZU  ' 

997758 

Z  .  1 
2-1 

008047 

8 

991953 

1013599485 

11 

50 

OJ7044 

9  r 

997745 

.  1 

21 

009298 

208 

990702  1 

10164:99482 

10 

51 

9.008278 

9/)K 

9.997732 

.  1 

2] 

9.010546 

9O7 

0.989454 

10192'99479 

9 

52 

009510 

zuo 
205 

997719 

.  1 

2  1 

011790 

207 

988210 

10221199476 

8 

53 

010737 

997706 

013031 

686969 

10250:99473 

7 

54 

011962 

204 
203 

997693 

2.1 

2  2 

014268 

206 

985732 

10279^99470 

6 

55 

013182 

203 

997680 

2  2 

015502 

_ 

984498 

10308,99467 

5 

56 

014400 

202 

997667 

9  9 

016732 

904 

983268  ! 

10337J99464 

4 

57 

015613 

9(  jo 

997654 

9  9 

017959 

ZUTC 

983041 

1036699461 

3 

58 

016824 

zuz 
om 

997U41 

Z  .  Z 
29 

019183 

.-•!q 

980817  |j  10395  99458 

2 

59 

018031  9", 

997628 

.  Z 
20 

020403 

0  Q 

9;9597  11)424199455 

1 

60 

019235  " 

997614 

.  ~f 

021620 

9783801  10453199452 

0 

Cosine,  j 

Sine. 

Cotang. 

Tang.   i  N.  cos.JN.^ine. 

~ 

84  D^rees. 

TABLE  11.     Log.  Sines  and  Tangents.  (6C)  Natural  Sines.           27 

' 

Sine. 

D.  10" 

Cosine.  ]D.  10" 

Tang.   iD.  10" 

Co  tang.   N.  sine. 

N.  cos.  j 

0 

9.019235 

Of\f\ 

9.997614  o  Q 

9.021620 

909 

10.978380  |  10453 

J9452  60 

1 

020435 

'200 

997601  J*J 

022834 

Z\j£ 
909 

977166  ;  10482 

99449  59 

2 

021632 

199 

1  QQ 

997588!  J'J 

024044 

Z\jZ 

201 

975956  ;  105  11 

9944(>  58 

3 

022825 

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Cosine. 

Sine. 

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Tang. 

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N.sine. 

/ 

83  Degrees. 

Log.  Sines  and  Tangents.  (7°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10' 

Cosine. 

D.  ](/ 

Tang. 

L>.  JO' 

Cotang.   :N.  sine 

N.  COH. 

0 

1 

9.085894 
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171 

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0 

Cosine. 

Sine. 

Cotang. 

Tang.    N.  C-OP. 

\.siiio. 

T" 

8:2  Degrees. 

TABLE  II.     Log,  Sines  and  Tangents.  (8°)  Natural  Sines,           29 

' 

Sine.' 

I).  10" 

Cosine. 

1).  li/' 

Tang. 

D.  10" 

Co  tang. 

N.  sine. 

N.  cos. 

o 

9.143555 

9.995753 

9.147803 

10.852197 

13917 

99027 

60 

1 

144453 

150 

995735 

3.0 

148718 

153 

851282!  13946 

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59 

2 

145349 

149 

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3.0 

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152 

850368  113975 

99019 

58 

3 

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149 

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150544 

152 

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99015 

67 

4 

147136 

149 

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848546 

14033 

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56 

5 

148026 

148 

995664 

3.0 

152363 

151 

847637 

14061 

99006 

55 

6 

148915 

148 

995646 

3.0 

153269 

151 

846731 

14090 

99002 

54 

7 

149802 

148 

995628 

3.0 

154174 

151 

845826 

14119 

98998 

53 

8 
9 

150686 
151569 

147 
147 

995610 
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3.0 
3.0 

155077 
155978 

150 
150 

844923 
844022 

14148 
14177 

98994 
98990 

52 
51 

10 

152451 

147 

995573 

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156877 

150 

843123 

14205 

98986 

50 

11 

9.153330 

147 

9.995555 

3.0 

9.157775 

150 

10.842225 

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149 

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47 

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145 
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31 

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145 

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35 

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145 

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34 

27 

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142 

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145 

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33 

28 

168008 

142 

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39 

172767 

145 

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32 

29 

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141 

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173634 

144 

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137 

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136 

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0 

Cosine. 

Sine. 

Cotang. 

Tang- 

N.  cos. 

N.sine. 

i 

81  Degrees. 

30          Log.  sines  and  Tangents.  (9°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10' 

Cotang. 

N.  sine. 

N.  cos.| 

o 

9.194332 

9.994620 

9.199713 

10.800287 

15643 

98769 

60 

1 

195129 

133 

994600 

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136 

799471 

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98764 

59 

2 

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133 

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136 
136 

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0 

Cosine. 

Sine. 

Cotang. 

Tang. 

IN.  cos. 

N.sine. 

~T~ 

80  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (10°)  Natural  Sines.          31 

/ 

Sine. 

D.  10"  Cosine. 

D.  10" 

Tang. 

D.  10"  Cotang. 

N.sine. 

N.  cos. 

0 

9.239670 

nq  9.993351 

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9.246319 

10.753681 

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98481 

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1 

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0 

Cosine. 

Sine. 

Co  tang. 

Tang. 

N.  eos. 

N.nne. 

' 

79  Degrees. 

32           Log.  Sines  and  Tangents.  (11°)  Natural  Sines.     TABLE  II. 

i  '     Sine.   1 

[>.  10" 

Cosine. 

U.  10" 

Tally;.   ID.  iu 

ixitang. 

N.  sine. 

N.  cos. 

0 

J.  280599 

"1  AQ 

9.991947 

41 

9.288652 

no 

10.711348 

19081 

98163 

60 

1 

281248 

lUo 

991922 

.  l 

289326 

z 

710674 

19109 

98157 

59 

2 

281897 

108 

991897 

4.1 

289999 

710031 

19138 

98152 

68 

3 

282544 

108 

991873 

4.  1 

290671 

709329 

19167 

98146 

57 

4 

283190 

108 

991848 

4.1 

291342 

708058 

19195 

98140 

56 

6 

28383f> 

108 

1  A  7 

991823 

4.  1 
41 

292013 

707987 

19224 

98135 

55 

6 

284480 

10  / 
1  n7 

991799 

.  i 
41 

292682 

707318 

19252 

J8129 

54 

7 

285124 

10  / 

1  A  7 

991774 

.  i 

A  9 

293350 

706650 

19281 

J8124 

53 

8 

285766 

ID/ 

991749 

4.  -^ 
,  o 

294017 

705983 

19309 

98118 

52 

9 

286408 

107 

991724 

4.2 

.  o 

294684 

111 

705316 

19338 

J8112 

51 

10 

287048 

107 

991699 

4.2 

295349 

111 

704651 

19366 

98107 

50 

1  1 

).  287687 

107 

3.  991674 

4.2 

9.296013 

111 

TO.  703987 

19395 

98101 

49 

12 

288326 

106 

991649 

4.2 
40 

296677 

111 
1  1  0 

703323 

19423 

98096 

48 

13 

288964 

106 

991624 

.  £ 

297339 

1  1U 

702661 

19452 

98090 

47 

14 

289600 

106 

991599 

4.2 
40 

298001 

110 
110 

701999 

19481 

98084 

46 

15 

290236 

106 

991574 

.* 

.  o 

298662 

1  1U 
1  1  A 

701338 

19509 

98079 

45 

16 

290870 

106 

991549 

4.2 

299322 

110 

700678 

19538 

98073 

44 

17 

291504 

106 

991524 

4.2 

299980 

110 

700020 

19566 

)8067 

43 

18 

292137 

105 

991498 

4.2 
40 

300638 

110 

1  OQ 

699362 

19595 

)8061 

42 

19 

292768 

105 

991473 

.* 

301295 

iuy 

698705 

19623 

98056 

41 

20 

293399 

105 

991448 

4.2 

A   O 

301951 

109 

1  AO 

698049 

19652 

98050 

40 

21 

9.294029 

105 

9.991422 

4.^ 
40 

9.302607 

iuy 

1  OQ 

10-697393 

19680 

98044 

39 

22 

294658 

105 

1  AT* 

991397 

.* 

40 

303261 

iuy 

1  AO 

696739 

19709 

98039 

38 

23 

295286 

ll/O 
1  (\A 

991372 

.<& 

40 

303914 

iuy 

1  AO 

696086 

19737 

98033 

37 

24 

295913 

104 

1  A  4 

991346 

.*> 

40 

304567 

1LM 
109 

695433 

19766 

98027 

36 

25 

296539 

104 

1  A/1 

991321 

.0 

40 

S05218 

lU^ 
108 

694782 

19794 

J8021 

35 

26 

297164 

104 

1  A/I 

991295 

.0 

40 

305869 

J.UO 
1  AQ 

694131 

19823 

38016 

34 

27 

297788 

104 

1  A/I 

991270 

.0 

40 

306519 

1U<^ 
1  08 

.  693481 

19851 

98010 

33 

28 

298412 

104 

1  f\  A 

991244 

.0 

40 

307168 

luo 

108 

692832 

19880 

98004 

32 

29 

299034 

104 

1  A4 

991218 

.0 

40 

307815 

iUO 
1  AQ 

692185 

19908 

97998 

31 

30 

299655 

104 

991193 

.0 

40 

308463 

1UO 

1  08 

691537 

19937 

97992 

30 

31 

9.300276 

103 

9.991167 

.0 

40 

9.309109 

1UO 
1  07 

10-690891 

19965 

97987 

29 

32 

300895 

103 

991141 

.0 

40 

309754 

1U  * 
1  07 

690246 

199&4 

97981 

28 

33 

301514 

103 

991115 

.O 

40 

310398 

1U  / 
1  r\7 

689602 

20022 

97975 

27 

34 

302132 

103 

991090 

;O 

40 

311042 

llf  I 
1  O"' 

688958 

2X)051 

97969 

26 

35 

302748 

103 

991064 

.0 

40 

311685 

1U< 
1  f>7 

688315 

20079 

97963 

25 

36 

303364 

103 

1  AO 

991038 

.O 
40 

312327 

1U< 
107 

687673 

20108 

97958 

24 

37 

303979 

WZ 

1  no 

991012 

.  o 

40 

312967 

lUi 

1  07 

687033 

20136 

97952 

23 

38 

304593 

l(}Z 
1  AO 

990986 

*  o 

A  Q 

313608 

i  \ft 
1  0fi 

686392 

20165 

97946 

22 

39 

305207 

l\)£ 
1  AO 

990960 

*1,  O 

A  Q 

314247 

AUO 

106 

685753 

20193 

97940 

21 

40 

305819 

\\K 

1  AO 

990934 

TC«  O 

4   A 

814885 

1UO 
106 

685115 

20222 

97934 

20 

41 

9.306430 

lUSe 

1  AO 

9  990908 

.  ^fc 

4   A 

9.315523 

1UD 

10I-! 

10-684477 

20250 

97928 

19 

42 

307041 

lift 

1  09 

990882 

•  ^* 

4  4 

316159 

J.UO 

106 

683841 

20279 

97922 

18 

43 

307650 

k\Ke 

990855 

4   A 

316795 

1  AM 

683205 

2030? 

97916 

17 

44 

308259 

101 

990829 

%*± 

4    A 

317430 

1UO 
1  Aii 

682570 

20336 

97910 

16 

45 

308867 

101 

1  O1 

990803 

•  ^ 

4  A 

318064 

1UO 
1  Or) 

681936 

20364 

97905 

15 

46 

309474 

1  Vrl 
1  A1 

990777 

.  *x 
A  A 

318697 

1UO 

105 

681303 

20393 

97899 

14 

47 

310080 

1U1 
1  A1 

990750 

'!•'* 
A  A 

319329 

i  n^ 

680671 

20421 

97893 

13 

48 

310685 

1O1 

1  A1 

990724 

T:  *  ^t 
A  A 

319961 

1UO 

1  0^ 

680039 

20450 

97887 

12 

49 

311289 

1UI 

990697 

4.  ^ 

320592 

JIUO 
1  AX 

679408 

20478 

97881 

11 

50 

311893 

100 

990671 

4.4 

321222 

lUo 

1  f\^ 

678778 

2050? 

97875 

10 

51 

9.312495 

100 
i  on 

9.990644 

4.4 
A  4 

9.321851 

lOo 
105 

10-678149 

20535 

97869 

9 

52 

313097 

1UU 

i  An 

990618 

rr  .  TC 
4-  4 

322479 

677521 

20563 

97863 

8 

63 

313698 

1UU 

i  on 

990591 

4t  .  ^ 

A  4 

323106 

676894 

20592 

97857 

7 

54 

314297 

1UU 

1  1  M  t 

990565 

'±.  ^t 

A   A 

323733 

104 

676267 

20620 

97851 

6 

55 

314897 

1UU 

i  t\i\ 

990538 

Q  .  Tt 
4  A 

324358 

lU^i 

1  04 

675642 

20649 

97845 

5 

56 

315495 

1UU 

1  AA 

990511 

•  TC 
4  K 

324983 

1UT: 
1  04 

675017 

2007? 

97839 

4 

57 

316092 

1UU 
QU 

990485 

^t  •  O 

d.  r> 

325607 

AUTC 

674393 

20706 

97833 

3 

56 

316689 

yy 

QU 

990458 

4  k  O 

A  ft 

326231 

673769 

20734 

97827 

2 

59 

317284 

yy 
ou 

990431 

4  .  O 
A  & 

326853 

1  O-t 

673147 

20763 

97821 

1 

60 

317879 

yy 

990404 

*±.  O 

327475 

lU-t 

672525 

20791 

97815 

0 

~CosiiieT~ 

Sine. 

Co  tang. 

Tang.   ||  N.  cos 

N.rfne 

' 

78  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (12°)  Natural  Sines. 

33 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.   jN  .  sine. 

i>,  CO*. 

0 

9.317879 

on  n 

9.990404 

4e 

9.327474 

1  no 

10.672526 

20791  97815 

60 

1 

318473 

yy  .  u 

98.8 

990378 

.0 

4  K, 

328095 

LUJ 
1  0*^ 

671905 

20820  97809 

59 

2 

319.W6 

f  kQ  *7 

990351 

4.  O 

328715 

LU<j 

671285 

20848 

97803 

58 

3 

319658 

9o  .  7 
98.6 

990324 

4.5 

4K 

329334 

103 
1  »•? 

6  1  0-Jou 

20877  97797 

57 

4 

320249 

no  A 

990297 

:  .  O 

329953 

i  j.> 

670U47 

20905 

97791 

56 

5 

320840 

9o.4 
98  3 

990270 

4.5 

4K 

330570 

103 

1  QQ 

669430 

20933  97784 

55 

6 

321430 

98  2 

990243 

.  o 

4K 

331187 

iUo 
i  no 

668813 

20962  97778 

54 

7 

322019 

oVo 

990215 

•  O 

4K 

331803 

lUo 
1  09 

668197 

20990  97772 

53 

8 

322607 

Jo  .  U 
07  q 

990188 

•  O 
4K 

332418 

1U4 

i  no 

667582 

2101997766 

52 

9 

323194 

J  I  .  y 

Q7  7 

990161 

.  O 
4K 

333033 

1U-* 
1  09 

666967 

2104797760 

51 

10 

323780 

y  i  .  i 
Q7  (\ 

990134 

.  O 
4K 

333646 

iinfl 

1  09 

666354 

2107697754 

50 

11 

9.324366 

J  i  .  O 

07  5 

9.990107 

.  O 

4fi 

9.334259 

L(J& 
i  no 

10.665741 

211049774S 

49 

12 

324950 

•J  i  .  O 

Q7  1 

990079 

:.  O 

4« 

334871 

1  \j£ 
1  AO 

665129 

2113297742 

48 

13 

325534 

y  /  .  o 

Q7  9 

990052 

•  .  o 

4f! 

335482 

IU-4 

i  no 

664518 

2116197735 

47 

14 

326117 

y  /  .  z 

O7  t\ 

990025 

.  O 

4(\ 

336093 

LU-6 

1  fkO 

663907 

2118997729 

46 

15 

326700 

y  i  •  \J 

Q.IA  Q 

989997 

.  o 

4c 

336702 

iUDI 

663298 

21218  97723 

45 

16 

327281 

yo  .  y 
96.8 

989970 

.  o 
4.6 

337311 

1  01 

662689 

2124697717 

44 

17 

327862 

Qfi  fi 

989942 

A   C 

337919 

iivl 

1  ni 

662081 

2127597711 

43 

18 

328442 

yo  .  o 

989915 

4  .  D 

4r» 

338527 

lUi 

661473 

21303  97705 

42 

19 

329021 

96.  5 

989387 

.O 

4£» 

339133 

101 

660867 

2133197698 

41 

20 

329599 

96.4 

9K  <2 

989860 

.0 

A  fi 

339739 

101 

660261 

2136097692 

40 

21 

9.330176 

o  .  -^ 

9.989832 

4  .  O 

4"  £! 

9.340344 

10.659656 

21388  97686 

39 

22 

330753 

96.  1 

989804 

.O 

340948 

101 

659052 

2141797680 

38 

23 

331329 

96.  0 

9~  Q 

989777 

4.  6 
4fj 

341552 

101 
i  nn 

658448 

2144597673 

37 

24 

331903 

O  .  u 

O~  ry 

989749 

.  o 

4ry 

342155 

1UU 

657845 

2147497667 

36 

25 

332478 

yo.  / 

QX  & 

989721 

.  1 

A  7 

342757 

100 

i  an 

657243 

2150297661 

35 

26 

333051 

yo  .  o 

QX  4 

989693 

TT.  1 

A  7 

343358 

1UU 
1  00 

656642 

2153097655 

34 

27 

333624 

yO  •  4: 

989665 

4.  i 

4rf 

343958 

\.\J\) 

656042 

21559 

97648 

33 

28 

334195 

95.3 

QX  O 

989637 

.7 

4rr 

344558 

100 

1  C\(\ 

655442 

21587:97642 

32 

29 

334766 

yo.  z 

QX  A 

989509 

.  1 

47 

345157 

1UU 
i  i\i\ 

654843 

21616!97636 

31 

30 

335337 

yj  .  \) 

Ql  Q 

989582 

.  t 

47 

345755 

1UU 
1  (\'\ 

654245 

21644  97630 

30 

i  31 

9.335906 

y-±.  y 

Q  1  W 

9.989553 

*  i 

A  7 

9.346353 

1U*J 
OQ  A 

10.653647 

2167297623 

29 

32 

336475 

y-±  .  o 

Q  1  M 

989525 

4.1 
4.  7 

346949 

\J  J  •  H 

QO  ^ 

653051 

2170197617 

28 

33 

337043 

y^t  .  o 

Q/1  X. 

989497 

4  .  / 

A  7 

347545 

yy  •  c: 

GO  O 

652455 

2172997611 

27 

34 

337610 

y4  .  o 

Q.1  /I 

989469 

4.  1 

A  7 

348141 

yy  .  >i 

GO  1 

651859 

2175897604 

26 

35 

338176 

y-i  .  4 

Q  1  Q 

989441 

4  .  / 

47 

348  ?35 

yy  .  i 

GO  f 

651265 

21786:97598 

25 

36 

338742 

y  ±  .  o 

uA  i 

989413 

.  / 

47 

349329 

yy  .  v 
98.  £ 

650671 

2181497592 

24 

37 

339306 

y4.  i 

989384 

.  * 

4rr 

349922 

9.j  17 

650078 

21843!97685 

23 

38 

339871 

94.  0 

no  (i 

989356 

.  / 

4rj 

350514 

o  .  * 
O-2  ^i 

649486 

21871  97579 

22 

39 

340434 

yo.y 

qq  7 

989328 

:.  / 

47 

351106 

yo  .  u 

QS  fi 

648894 

21899  97573 

21 

40 

340996 

yo.  t 

QO   C- 

989300 

.  / 

47 

361697 

C7O  .  O 

UW  Q 

648303 

21928  97566 

20 

41 

9.341558 

yo  .  o 

nQ  Pi 

9.989271 

.  / 

47 

9.352287 

yo  .  o 
*IP  9 

10.647713 

21956  97560 

19 

42 

342119 

yo  .  O 
nQ  A 

989243 

.  / 

A  7 

352876 

i7O  .  A 
QL?   I 

647124 

21985  97553 

18 

43 
44 

342679 
343239 

\jo  .  4 
93.2 

f\Q  1 

989214 
989186 

4  .  / 

4.7 

4r* 

353465 
354053 

yo  .  i 
98.0 
07  o 

646535 
645947 

22013  97547 
22041197541 

17 

16 

45 

343797 

9<J-  1 

989157 

.  / 

4.7 

354640 

y  /  .  y 

()-  7 

645360 

22070 

97534 

15 

46 

344355 

93  .  0 

rkO  Q 

989128 

.  / 

A  ft 

355227 

%M  *  I 

07  « 

644773  22098 

97528 

14 

47 

48 

344912 
345469 

9-^.y 

92.7 
Q9  fi 

989100 
989071 

4.o 

4.8 

4C 

355813 
356398 

y  /  .  o 
97.5 

C)7  ,d 

644187  |  22126 
643^02  22155 

97521 
97515 

13 
12 

49 

346024 

y^  .  o 

fiO  K 

989042 

.  O 
A  Q 

356982 

y  <  .  T 

07  q 

643018  '22183 

97508 

11 

50 

346579 

y^.  o 

nO  /I 

989014 

4  .  O 

4« 

357566 

y  *  .  o 
Q7  1 

642434 

22212 

97502 

10 

51 
52 

9.347134 

347687 

y/*  .  4 
92.2 
/v~>  -i 

9.988985 
988956 

.  O 

4.8 

4U 

9.358149 
358731 

y  /  .  i 

97.0 

OH  Q 

10.641851  22240 
641269)  122268 

97496 

97489 

9 

8 

53 

348240 

92.1 

988927 

.  O 

40 

359313 

yo  .  y 

or;  Q 

640687 

22297 

97483 

7 

54 

34879-2 

92.  0 

988898 

.  o 

4Q 

359893 

yu  .  o 

QO  n 

640107 

3232597470 

6 

55 
56 
57 

349343 
349893 
350443 

91  .9 
yl.7 
91.6 

988869 
988840 
988811 

.0 

4.8 

4.8 
40 

360474 
361053 
361632 

y  o  ,  / 
96.6 
96.5 

Q£;  r 

639526 
638947 
638368 

22353 

|  22382 
22410 

97470 
97463 
97457 

5 
4 
3 

68 

350992 

91.5 

988782 

.y 

4n 

362210 

yo.o 

f\f\  o 

637790 

22438 

97450 

2 

59 

351540 

91  .4 

988753 

.9 

362787 

yo.z 

ot;  1 

637213 

22467 

97444 

1 

60 

352088 

91.3 

988724 

4.9 

363364 

yt>.  i 

636636 

22495 

97437 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

N.sine. 

77  Degrees. 

34          Log.  Sines  and  Tangents.  (13°)  Natural  Sines.     TABLE  11. 

~i 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.  i  N.sine_ 

N.  cos. 

0 
1 

9.352088 
352635 

91.1 

9.988724 
988695 

4.9 
4f\ 

9.363364 
363940 

96.0 

(\K.  O 

10.636636 
636060 

22495 
22523 

97437 
97430 

60 
59 

2 

353181 

91  .0 

988666 

.y 

4Q 

364515 

y£>.y 

f^K.  O 

635485 

22552 

97424 

58 

3 

353726 

90.9 

988636 

.y 
4(\ 

365090 

95.0 

r\K.  >7 

634910 

22580 

97417 

67 

4 

354271 

90.8 

988607 

.y 

365664 

95.7 

634336 

22608 

97411 

66 

5 

354815 

90.7 

988578 

4.9 

366237 

95.5 

633763 

22637 

97404 

65 

6 

355358 

90.5 

988548 

4.9 

366810 

95.4 

633190 

22665 

97398 

54 

7 

355901 

90.4 

988519 

4.9 
40 

367382 

95.3 

632618 

22693 

97391 

53 

8 

356443 

90.3 

988489 

.y 

367953 

95.2 

632047 

22722 

97384 

52 

9 

356984 

90.2 

988460 

4.9 
40 

368524 

95.1 

631476 

22750 

97378 

61 

10 

357524 

90.  1 

988430 

.y 

4Q 

369094 

95.0 

630906 

2277897371 

60 

11 

9.358064 

89.9 

9.988401 

.y 

4Q 

9.369663 

94.9 

f\A   Q 

10.630337 

2280797365 

49 

12 

358603 

89.8 

988371 

.y 

4Q 

370232 

94.8 

629768 

2283597358 

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TABLE  II.     Log.  Sines  and  Tangents.  (14°)  Natural  Sines.          35 

Sine. 

D.  10' 

Cosine. 

D.  10"   Tang. 

D.  10' 

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0 

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i 

75  Degrees. 

36          Log.  Sines  and  Tangents.  (15°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

I).  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

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0 

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59 

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net  x 

642981 

27536 

96134 

1 

60 

440338 

73.5 

982842 

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542504 

27564 

96126 

0 

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Sine. 

Cotang. 

Tang. 

N.  cos.  |N.  sine. 

/ 

74  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (16°)  Natural  Sines.          37 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang. 

N.  sine. 

N.  cos.j 

0 

9.4t0338 

70  A 

9.982842 

6O 

9.457496 

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10.542504 

27564 

96126 

60 

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440778 

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982805 

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457973 

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542027 

27592 

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59 

2 

441218 

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982769 

6.0 

458449 

79.3 

541551 

27620 

96110 

58 

3 

441658 

73  .2 

982733 

6.  . 

458925 

79  .3 

541075 

27648 

96102 

57 

4 

442096 

73  •  1 

982696 

6. 

459400 

79.2 

540600 

27676 

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5 

442535 

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982660 

6. 

459875 

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540125 

27704 

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6 

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to  .  u 

982624 

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460349 

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539651 

27731 

96078 

54 

7 

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8 

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461297 

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27787 

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52 

9 

444284 

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461770 

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61 

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1 

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6.4 

485339 

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29247 

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0 

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Sine. 

Cotang. 

Tang. 

N.  cos. 

N.sine. 

' 

73  Degrees. 

Log.  Sines  and  Tangents.  (17°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10' 

Cosine. 

D.  10 

Tang. 

D.  10' 

Cotang.   N.  sine 

N.cos 

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9.485339 

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10.514661 

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493407 

30570 

95213 

12 

49 

485682 

65.5 

978655 

.  O 

6Q 

607027 

m  o 

492973 

3059? 

95204 

n 

50 

486075 

65.5 

978615 

.0 

507460 

72.2 

492540 

30625 

95196 

10 

61 

9.486467 

65.4 

9.978574  "'S 

9.607893 

wo  1 

0.492107 

30653 

95186 

9 

52 

486860 

66.3 

978533 

U  .0 

60 

608326 

72.  1 

491674 

30680 

95177 

8 

53 

487251 

65.3 

978493 

,0 
6Q 

508759 

72.  1 

491241 

30708 

95168 

7 

54 

487643 

65.2 

978452 

.  0 

bo 

609191 

72.0 

wi   O 

490809 

30736 

95159 

6 

55 

488034 

65.1 

978411 

.  O 
6Q 

609622 

71  .y 

wi  n 

490378 

30763 

95150 

6 

66 

488424 

65.1 

•978370 

.0 
6O 

610054 

71  .y 

W1   Q 

489946 

30791 

95142 

4 

57 

488814 

65.0 

978329 

.O 
6Q 

510485 

71.0 

489515 

30819 

95133 

3 

58 

489204 

65.0 

9  78288 

.  0 
6Q 

610916 

71  .8 

489084 

30846 

95124 

2 

59 

489593 

34.  y 

978247 

.  o 

60 

511346 

71.7 

w-l   £? 

488654 

30874 

95115 

1 

60 

489982 

64.8 

978206 

.O 

511776 

71  .0 

488224 

3090'^ 

95106 

0 

Cos[ne7~ 

Sine. 

Cotang. 

Tang.   ||N.  cos. 

N.sinc. 

~~r 

T>  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (18°)  Natural  Sines.          39 

_       ._ 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.   N.  sine  N.  cos. 

0 

9.489982 

RA.  R 

9.978206 

60 

9.511776 

71  R 

10.488224  30902 

95106 

60 

1 

490371 

D4.  o 

C1   Q 

978165 

.  o 

60 

512206 

<  1  .D 

rT  1   £» 

487794!!  30929 

95097 

59 

o 

490759 

O1!  •  o 

978124 

•  o 

60 

612635 

/  1  .D 

487365  |  i30957 

95088 

58 

3 

491147 

64.7 

978083 

.0 

613064 

71  .5 

486936  130985 

95079 

57 

4 

491535 

64.6 

P.A  P. 

978042 

6.9 

6Q 

613493 

71.4 

IJ-I    A 

486507  i  31012 

95070 

66 

6 

491922 

D4.D 

64  5 

978001 

.y 

6  a 

613921 

/1  .4 

71   0 

486079  31040 

95061 

55 

6 

492308 

64  4 

977959 

•  y 

60 

514349 

4  1  .  O 

71  °i 

485651  31068 

95052 

54 

7 

492695 

O4:  .  *± 

64  4 

977918 

.  y 

6q 

614T77 

4  1  .  O 

71  0 

485223  31095 

95043 

53 

8 

493081 

DHt.  *± 

64  ^ 

977877 

.  y 

6q 

615204 

i  x  «2p 

71  0 

484796 

31123 

95033 

52 

9 

493466 

O^.  O 

R4  2 

977835 

•  y 
6  9 

615631 

1  A  •  « 

71  1 

484369 

31151 

95024 

51 

10 

493851 

O^T  .  Z 

fi4  1 

977794 

6q 

616057 

•  •*.  I 

71  A 

483943 

31178 

96016 

50 

11 

9.494236 

D'i.  Z 

64  1 

9.977752 

.  y 
6  a 

9.516484 

i  A  .  U 

71  H 

10.483516 

31206 

95006 

49 

12 

494621 

O*x.  1 

64  1 

977711 

.  y 

6q 

516910 

/  1  .  U 
70  Q 

483090 

31233 

94997 

48 

13 

495005 

D^.  1 

fi.4  0 

977669 

.  y 

6q 

617335 

iv  .  y 

70  Q 

482665 

31261 

94988 

47 

14 

495388 

O^.  \j 
f\\  Q 

977628 

.  y 
6q 

617761 

/  u.  y 

70  ft 

482239  31289 

94979 

46 

15 

495772 

DO  .  y 
cq  Q 

977686 

.  y 
6q 

518185 

/u  .  o 

70  ft 

481816 

31316 

94970 

45 

16 

496154 

DO  .  y 
f\\  P. 

977644 

.  y 

7  0 

618610 

i  U  .  o 
70  7 

481390 

31344 

94961 

44 

17 

496537 

DO  .  o 

CO  fi 

977503 

i  ,  \) 

7  0 

619034 

/U.  7 
70  ft 

480966 

31372 

94952 

43 

18 

496919 

Do.  / 

CO   7 

977461 

i  .  U 

7  0 

519458 

/U.O 
70  P. 

480542 

31399 

94943 

42 

19 

497301 

DO  .  / 

CO  C. 

977419 

•  •  v 

7  n 

519882 

/U.  b 

*7A  K 

480118 

31427 

94933 

41 

20 

497682 

Do.D 

977377 

i  .  U 

7r\ 

520305 

/U.  5 

479695 

31454 

94924 

40 

21 

9.498064 

63.6 

CO  C 

9.977335 

.0 

7  0 

9.520728 

70.5, 
70  A 

10.479272 

3148294915 

39 

22 
23 
24 
25 
26 

498444 
498825 
499204 
499584 
499963 

Do  .  O 

63.4 
63.4 
63.3 
63.2 

cq  9 

977293 
977251 
977209 
977167 
977125 

'  .  v 

7.0 
7.0 
7.0 
7.0 

7  0 

521151 
621573 
621995 
622417 
622838 

/  U  4 

70.3 
70.3 
70  3 
70.2 

rrf\  r> 

478849 
478427 
478005 
477583 
477162 

31510!94906 
3153794897 
3156594888 
3159394878 
3162094869 

38 
37 
36 
35 
34 

27 

600342 

DO  .  Z 
cq  i 

977083 

/  .  U 

7  0 

623259 

/U.  2 

r-rf\  1 

476741 

31648 

94860 

33 

28 

500721 

Do  .  1 

CO   1 

977041 

/  .  U 

7  n 

523680 

'".  1 

r~i\  -t 

476320 

31675 

94851 

32 

29 

501099 

Do  ,  1 

976999 

i  .U 

524100 

/U.I 

475900 

31703 

94842 

31 

30 

601476 

63  .  0 

976957 

7.0 

624520 

70.0 

475480 

31730^94832 

30 

31 

9.501854 

62.9 

9.976914 

7.0 

9.524939 

69.9 

tO-  475061 

31758^94823 

29 

32 
33 
34 
35 

502231 
502607 
602984 
503360 

62.9 
62.8 
62.8 
62.7 

R9  fi 

976872 
976830 
976787 
976745 

7.0 
7.1 
7.1 
7.1 

7  1 

525359 
625778 
626197 
626615 

69.9 
69.8 
69.8 
69.7 

£?Q  i*r 

474641 

474222 
473803 
473385 

3178694814 
31813948U5 
3184194795 
31868194786 

28 
27 
26 
25 

36 
37 

603735 
504110 

D-*  .  D 

62.6 

976702 
976660 

/  ,  1 

7.1 

7-t 

627033 
527451 

oy  ,  / 
69.6 

472967 
472549 

31896  94777 
3192394768 

24 
23 

38 

504485 

62.5 

976617 

.  l 

71 

627868 

69.6 

472132 

3195194758 

22 

39 

604860 

62.5 

976574 

.1 

628285 

69.5 

471715 

3197994749 

21 

40 

605234 

62.4 

976532 

7.1 

528702 

69.5 

471298 

32006  94740 

20 

41 

9.505608 

62.3 

r»o  Q 

9.976489 

7.1 

71 

9.629119 

69.4 

r>(\  ci 

10.470881 

32034^94730 

19 

42 

505981 

O-4  .0 

fi9  "2 

976446 

.  i 

7  1 

629535 

t>y  .  o 

fiQ  °. 

470465 

3206194721 

18 

43 

506354 

u  -  •  ~ 

c.o  o 

976404 

i  .  A 

7  1 

529950 

Dy  ,  o 

f\Q  Q 

470050 

3208994712 

17 

44 

506727 

D.4  .  ^ 

976361 

•  •* 
71 

630366 

Dy  .  o 

469634 

32116'94702 

16 

45 

607099 

62  .  1 

976318 

.  1 
71 

630781 

69  .2 

469219 

32144'94693 

15 

46 

507471 

62.  0 

976275 

.  1 
71 

531196 

69.1 

468804 

3217194684 

14 

47 

507843 

62.0 

976232 

.  1 

70 

531611 

69.1 

468389 

32199  94674 

13 

48 

608214 

61  .9 

976189 

,Z 

7'  O 

632025 

69.0 

467975 

32227  94665 

12 

49 

508585 

61  .9 
fil  H 

976146 

.^ 
7  2 

532439 

69.0 
fitf  q 

467561 

32250  94656 

11 

50 

508956 

Dl  .  o 

C1   O 

976103 

/  .  ^ 

7  2 

532853 

DO  .  y 
fiQ.  q 

467147 

32282  94646 

10 

51 

9.509326 

D  1  .  o 
P.1  7 

9.97G060 

*  .  •* 

7  2 

9.633266 

DO  .  y 

fiK  ft 

10.466734 

32309  94637 

9 

52 

509696 

Dl  .  ' 

P.1  fi 

976017 

i  .  -^ 
7  1 

533679 

Do  ,  o 

fiR  C 

466321 

32337  94627 

8 

53 

610065 

Dl  .O 

975974 

f  .  ~ 
70 

534092 

Do  .  o 

/»c  * 

465908 

32364  94618 

7 

54 

610434 

61  .6 

975930 

.  Z 
70 

634504 

DO.  7 

465496 

32392 

94609 

6 

55 

610803 

61.5 

975887 

.  •* 
70 

534916 

68  .  7 

465084 

32419 

94599 

5 

56 

511172 

61  .5 

975844 

.* 

635328 

68.6 

464672 

32447 

94590 

4 

57 

611540 

61  .4 

976800 

7.2 

635739 

68.6 

464261 

32474  94580 

3 

68 

511907 

61  .3 

975757 

7.2 

636150 

68.5 

463850 

3250294571 

2 

59 

512275 

61  .3 

975714 

7.2 

636561 

68.6 

463439 

3252994561 

1 

60 

512642 

61  .2 

975670 

7.2 

536972 

68.4 

463028 

3255  ,  94552 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  coR.JN.sine. 

T 

71  Degrees. 

40           Log.  Sines  and  Tangents.  (19°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

U.  10 

Cosine. 

i).  iU 

Tang. 

i>.  10"   Cotaug. 

N.  sine.|N.  cos.| 

o 

9.512642 

9.975670 

70 

9.536972 

rjo  A 

10.463028 

3255 

94552 

60 

513IM 

61  .2 
n  i 

9756-27 

.  <J 

7  ^ 

537382 

DO  .^ 
CQ  q 

462618 

3258 

94542 

59 

t 

513375 

ol  .  l 

975583 

/  .  o 

7  ^ 

537792 

DO  .  i 

r>Q  * 

462208 

3261 

94533 

58 

j 

513741 

ol  .  1 

975539 

/  .  o 

538202 

bo  .  i 
c-0  9 

461798 

3263 

94523 

57 

i 

514107 

61  .( 

97549( 

79 

638611 

bo  .  z 
fiP.  9 

461389 

3266 

94514 

56 

5 

614472 

60  *• 

975452 

7  3 

539020 

bo  .Z 
68  1 

460980 

3269 

94504 

55 

6 

514837 

cf\  £ 

975408 

70 

539429 

bo  .  . 

460571 

3272 

94495 

54 

515202 

bU.c 

fin  s 

975365 

.  O 

7  3 

639837 

r«  i 

460163 

3274 

94485 

53 

8 

615586 

bu  •  c 

975321 

7  3 

640245 

o' 

459755 

3277 

94476 

52 

c 

615930 

fin  r 

975277 

7  ^ 

640653 

R7  t 

459347 

3280 

94466 

51 

10 

516294 

bU.  / 

975233 

1  .  O 

7  3 

641061 

b  i  .  J 

458939 

3283 

94457 

60 

11 

9.516657 

PA  P. 

9.975189 

I  .  O 

9.641468 

R7  £ 

10.458532 

3285 

94447 

49 

12 

617020 

bU  -0 

975145 

7  ^ 

641876 

b/  .0 
R7  P. 

458125 

3288 

94438 

48 

13 

517382 

AX 

975101 

/  .  o 
70 

542281 

b/  .0 

457719 

3291 

94428 

47 

14 

517745 

60  .  4 

975057 

•  o 

542688 

fi7  *- 

457312 

3294 

94418 

46 

15 

518107 

rn  r 

975013 

7q 

643094 

D  /  .  i 
R7  f 

456906 

32969 

94409 

45 

16 

618468 

«A'C 

974969 

.  «J 

7  4 

643499 

D  /  .  1 

S7  ( 

456501 

3299 

94399 

44 

17 

618829 

fin  9 

974925 

i  .  *x 

7  4 

643905 

R7  t 

456095 

33024 

94390 

43 

18 

519190 

bO-z 

974880 

i.4 

7   A 

644310 

0  i  .  C 
/-jO"  p- 

455690 

3305 

94380 

42 

19 

519551 

60-  1 

orv  -I 

974836 

.4 

7   A 

644715 

a/  .0 

455285 

33079 

94370 

41 

20 

519911 

bU-  1 

974792 

.  4 

7  4 

545119 

37  / 

454881 

33106 

94361 

40 

21 

9.520271 

oU.U 

9.974748 

9.545524 

3  /  .  T 

10.454476 

33134 

94351 

39 

22 

620631 

60.0 
59  9 

974703 

7.4 
7  4 

645928 

37.  c 

R7  c 

454072 

3316 

94342 

38 

23 

520990 

K,q  q 

974659 

1  ,*r 

7  4 

646331 

3  /  .  c 

R7  Q 

453669 

33189 

94332 

37 

24 

521349 

oy  .  y 

974614 

/.4 

7  4 

646735 

3  i  .  •« 
fi7  2 

453265 

33216 

94322 

36 

25 

621707 

Kf\  Q 

974570 

1.4 

647138 

3  /  .^ 
a<*f  1 

452862 

33244 

94313 

35 

26 

622066 

a9.o 

974525 

7.  4 

647640 

67.1 

/.-y   -I 

452460 

33271 

94303 

34* 

27 

522424 

69.7 

974481 

7.4 

647943 

67.1 

an  A 

452057 

33298 

94293 

33 

28 

622781 

59.6 

974436 

7.4 

548345 

37.1 

451655 

33326 

94284 

32 

29 

623138 

59-6 

974391 

7.4 

7  4 

648747 

ifi  <H 

451253 

33353 

94274 

31 

30 

523495 

c 

974347 

i  .  4 

7er 

549149 

bb  .  y 

r;/j  Q 

450851 

33381 

94264 

30 

31 

9.523852 

a9.5 

^Q  A 

9.974302 

.6 

7  fi 

9.649550 

bb.y 
Rfi  8 

10.450450 

33408 

94264 

29 

32 

524208 

oy  .4 

974257 

/  .  0 

649951 

)b  .  o 

r*r>  Q 

450049 

33436 

94245 

28 

33 
34 

524564 
624920 

59.4 
69.3 

974212 
974167 

7.  5 
7.5 

650352 
650752 

bb.o 
66.7 

/?/-.  ry 

449648 
449248 

33463 
33490 

94235 

94225 

27 
26 

35 

625275 

59.3 

974122 

7.6 

651152 

bb.  i 

£»/>  y-> 

448848 

33518 

J4215 

25 

36 

625630 

59.2 

974077 

7.5 

651552 

)6.b 

448448 

33545 

94206 

24 

37 

625984 

59-1 

974032 

7.6 

651952 

56.6 

448048 

33573 

94196 

23 

38 

626339 

59-1 

973987 

7.5 

652351 

56.5 

447649 

33600 

4186 

22 

39 

626693 

39-0 

973942 

7.6 

652750 

66.5 

T>  K. 

447250 

33627 

4176 

21 

40 

627046 

59  •  0 

co  n 

973897 

7.5 

553149 

>b.o 

446851 

33655 

4167 

20 

41 

9.527400 

X5-9 

9.973852 

7.5 

9.563548 

66.4 

0.446452 

33682 

4157 

19 

42 

627753 

>8-9 

CO  Q 

973807 

7.5 

653946 

66.4 

446054 

33710 

4147 

18 

43 

628105 

>o.o 

973761 

7.5 

654344 

66.3 

445656 

33737 

4137 

17 

44 

528458 

)8-8 

973716 

7.5 

654741 

66.3 

445259 

33764 

4127 

16 

45 
1  46 

528810 
529161 

S8-7 
58  7 

~O   f» 

973671 
973625 

7.6 
7.6 

555139 
655536 

66.2 
66.2 

444861 
444464 

33792 
33819 

4118 
4108 

5 
14 

47 

529513 

oo  -b 

973580 

7.6 

666933 

66.1 

444067  !  33846 

4098 

3 

48 

529864 

)8-6 

973535 

7.6 

656329 

66.1 

443671  1 

33874 

4083 

2 

49 

530215 

>8-5 

973489 

7.6 

556725 

66.0 

443275  1 

33901 

4078 

1 

50 

530565 

S8-5 

973444 

7.6 

557121 

66.0 

442879  !  33929 

4068 

0 

51 

52 

.530915 
531265 

58-4 
58-4 

-Q  q 

.973398 
973352 

7.6 

7.6 
7(? 

.557517 
557913 

65.9 
65.9 

fi^  Q 

0.442483  3395b 

442087  il  33983 

4058 
4049 

9 

8 

53 

531614 

3o  •  O 

973307 

.b 

558308 

30  .  y 

441692  34011 

4039 

7 

54 
55 

531963 
532312 

58-2 
58-2 

973261 
973215 

7.6 
7.6 

558702 
659097 

65.8 
65.8 

441298  !i  34038 
440903  34065 

4029 
4019 

5 

56 

532661 

68.  1 

973169 

7.6 

559491 

65.7 

440509 

34093 

4009 

4 

57 

533009 

58.1 

973124 

7.6 

559885 

65.7 

440115 

34120 

3999 

3 

58 

53335  / 

58.0 

7.6 

560279 

65.6 

439721  ! 

34147 

3989 

2 

'  59 

533704 

!>8  0 

973032 

7.6 

560673 

35.6 

439327!  134175 

3979 

1 

60 

534052 

)7.9 

972986 

7.7 

561066 

66.5 

438934 

34202 

3969 

o 

Cosine. 

Sine. 

Co  tang. 

Tang. 

N.  cos. 

.sine. 

70  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (20°)  Natural  Sines.          41 

' 

Sine. 

D.  10' 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotanir.   N.  sine. 

N.  cos. 

0 

9.534052 

57  8 

9.972986 

7  7 

9.561066 

65  5 

10.438934!  34202 

93969 

60 

1 

534399 

972940 

i  •  i 

661459 

438541|  134229 

93959 

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8.  1 

584177 

62.9 

415823 

35837 

93358 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

X.sino. 

' 

69  Degrees. 

42          Log.  Sines  and  Tangents.  (21°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10' 

Cosine. 

D.  10' 

Tang. 

D.  10"i  Cotang.  jjN.sine. 

N.  cos.  - 

0 

9.554329 

X,1   Q 

9.970152 

9.584177 

71710.415823" 

35837 

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1 

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584555 

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89  Q 

415445 

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9 

554987 

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970055 

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684932 

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GO  Q 

415068 

35891 

93337  58 

3 

555315 

54.7 

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685309 

2  .  o 

GO  Q 

414691 

35918 

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4 

555643 

54.7 

969957 

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585686 

2.0 

414^14 

35945 

933  16  1  5;i 

6 

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64.6 

969909 

8. 

686062 

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413938 

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6 

7 

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8. 

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r>f\  rt 

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2 

59 

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(  06046 

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1 

60 

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8  .  6 

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37461 

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0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

N.pinc. 

~T~ 

68  Degrees. 

TABLE  IT.     Log.  Sines  and  Tangents.  (22°)  Natural  Sines.          43 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotaug.  iN.sinc. 

N.  cos. 

0 

9.573575 

52  1 

9.967166 

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10.389603 

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6 

55 

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49  8 

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CO  rj 

373907 

3893992107 

6 

56 
57 

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590984 

49!7 

49  7 

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373555 
373203 

38966  192096 
38993192085 

4 

3  ; 

58 
59 
60 

591282 
591580 
591878 

49.7 
49.6 

964133 
964080 
964026 

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8.9 
8.9 

627149 
627501 

627852 

OO.  D 

58.6 
58.5 

372851 
372499 
372148 

39020!92073 
3904692062 
39073|92050 

2 

1 
0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cosJN.sine. 

' 

67  Degrees. 

Log.  Sinos  and  Tangents.  (23°)  Natural  Sines.     TABLE  II. 

' 

Sin;;. 

1).  lu' 

Cosine. 

U.  1U 

Tang. 

D.  Ju 

Cotang.  ;  N.  sine 

N.  cos 

0 

9.591878 

49  6 

9.964026 

8  Q 

9.627852 

58  5 

10.372148  39073 

92050 

60 

1 

592176 

49  a 

963972 

o  .  J 

8Q 

628203 

58 

371797 

39100 

92039 

59 

r 

592473 

49  5 

963919 

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8  9 

628554 

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371446 

39127 

92028 

58 

3 

592770 

963865 

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628905 

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co 

371095  39153 

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57 

4 

593067 

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963811 

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629255 

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370745  i  39180 

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56 

5 

593363 

4:y  .  4 

4-9  4 

963757 

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Q  0 

629606 

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CO 

370394  39207 

91994 

55 

6 

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49  1 

963704 

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Q  f) 

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54 

7 

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963650 

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919^6 

50 

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9.595137 

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10.368296  ;39367 

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j  39394 

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48 

13 

595727 

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632401 

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598021 

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632750 

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58.1 

367250 

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963217 

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597490 

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55 

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56 

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47  4 

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352778 

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4 

57 

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4  /  .  4: 

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9  3 

647562 

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16  7 

352438 

40594 

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3 

58 

608745 

47  3 

960843 

94 

647903 

Ou  .  / 

Sfi  7 

352097 

40621 

)1378 

2 

59 

609029 

960786 

648243 

JO  .  / 

351757 

40647 

H366 

1 

60 

609313 

47.3 

9J0730 

9.4 

648583 

>6.7 

351417 

40674 

31355 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

X.sine. 

' 

'                             C6  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (24°)  Natural  Sines.          45 

' 

Sine. 

D.  10 

Cosine. 

D.  10' 

Tang. 

D.  10 

Cotang.  j  N.  sine 

N.  cos 

0 

9.609313 

47  ' 

9.960730 

9   A 

9.648583 

^ 

10.351417 

40674 

91355 

60 

1 

609597 

4  1  .  t 
47  1 

960874 

.  *± 

9   A 

648923 

06  .  ( 

351077 

40700 

91343 

59 

r 

609830 

4  /  .  ^ 

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649263 

Xfi  f. 

350737 

4072, 

91331 

58 

« 

610164 

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960561 

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oo  .  o 

350398 

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91319 

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610447 

47.2 

47  1 

960505 

9.4 
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649942 

56  6 

350058 

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610729 

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4  7  1 

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94 

650281 

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91260 

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4  i  .  I 

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10.347688 

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347012 

41019 

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47 

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346674 

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56.1 

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43 
42 

19 
20 

614665 
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56.1 

344989  ''41178  91128 

344652  I  41204  '91  lib 

41 
40 

21 

9.615223 

46  .  5 

AG  X 

9.959539 

9.5 

9K 

9.655684 

56.1 

10.344316  i  41231  91104 

o9 

22 

615502 

40  .  C 

959482 

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656020 

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343980  j:  41  257  9  1092 

38 

23 

615781 

46.5 

959425 

9.5 

656356 

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343644  4128491080 

37 

24 

616030 

46.4 

959368 

9.5 

656692 

56.0 

343308 

14131091068 

36 

25 
26 

616338 
616616 

46.4 
46.4 

959310 
959253 

9.5 
9.6 

657028 
657364 

55.9 
55.9 

342972 
342636 

41337|91056 
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35 
34 

27 

616894 

48.3 

9591  95 

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657699 

55.9 

342301 

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33 

28 

617172 

46.  3 

959138 

H  '  ~   658034 

55.9 

341966 

4141691020 

32 

29 
30 
31 
32 

617450 
617727 
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618281 

46.2 
46.2 
46.2 
46.1 

959081 
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9.958965 

958908 

9.6 
9.6 
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9.6 

658369 
658704 
9.659039 
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55.8 
55.8 
55.8 

55.8 

341631 
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41443  (91  008 
4146900996 

4149690984 
4152290972 

31 
30 

29 
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618558 

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958850 

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55  .  7 

340292 

4154990960 

27 

34 

618834 

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958792 

9.6 

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55.7 

339958 

41575  90948 

26 

35 

619110 

46.0 
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958734 

9.6 

660376 

55.7 

339624 

41602  90936 

25 

36 

619386 

40  .  U 

958677 

9.6 

660710 

55  .  7 

339290 

41628  90924 

24 

37 

619662 

46.  0 

958619 

9.6 

661043 

55.6 

338957 

41655  190911 

23 

38 

619938 

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958561 

9.6. 

661377 

55.6 

338623 

4168190899 

22 

39 

620213 

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958503 

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661710 

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338290 

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21 

40 

620488 

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958445 

9.7 

662043 

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20 

41 

9.620763 

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9.958387 

9.7 

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55.5 

0  337624 

4176090863 

19 

42 
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958329 

968271 

9.7 

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662709 
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337291 
336958 

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41813'90839 

18 
17 

44 

621587 

4o.  7 

A  X  rj 

958213 

9.7 

663375 

55.4 

336625 

41840^0826 

16 

45 
46 

621861 
622135 

4o.7 
45.6 

958154 
968096 

9.7 

9.7 

663707 
664039 

55.4 
55.4 

336293 
335961 

4186690814 
4189290802 

15 
14 

47 

622409 

r? 

958038 

9.7 

664371 

35.3 

335629 

4191990790 

13 

48 

622682 

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957979 

9.7 

664703 

55.3 

335297 

41945  90778 

12 

49 

622956 

4O  .  0 

A  X  x 

957921 

9.7 

665035 

35.3 

334966 

4197290766 

11 

50 
51 

623229 
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4O.  0 

45.5 

957863 
J.  957804 

9.7 
9.7 

665366 
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55.3 
55.2 

334634 
0.334303 

4199890753 
4202490741 

10 
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52 
53 
54 

623774 
624047 
624319 

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45.4 
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4**  o 

957746 

957687 
957628 

9.7 
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666029 
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55.2 
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333971 
333620 
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4205190729 
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957511 

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667352 

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332648 

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4 

57 

625135 

46.3 

957452 

9.8 

667682 

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332318 

42183  i 

>0668 

3 

58 

625406 

15.2 

AK.  O 

957393 

9.8 

668013 

35.0 

331987 

42209  i 

)0655 

2 

59 

625677 

10  .2 

957335 

J.8 

668343 

35.  0 

331657 

42235  £ 

10643 

1 

60 

625948 

15.2 

957276 

9.8 

668672 

o5.0 

331328 

42262  £ 

10631 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos.lk.sine. 

' 

65  Degrees. 

•   4G          Log.  Sines  and  Tangents.  (25°)  Natural  Sines.     TABLE  II. 

Sine. 

D.  10' 

Cosine.  }D.  10" 

Tang. 

D.  10' 

Cotang.  |,N.sine. 

N.  cos. 

0 

9.  625948 

AK.  1 

9.957276 

90 

9.668673 

CK  0 

10.331327 

42262 

90631 

60 

1 
2 

626219 
626490 

40  .  1 

45.1 

A  X   1 

957217 
957158 

.  o 
9.8 

90 

669002 
669332 

OO  .  U 

54.9 
Mq 

330998 
330868 

42288 
42315 

90613 
90606 

59 

68 

3 

626760 

40  .  1 
4C  A 

957099 

.  o 

9Q 

669661 

.  y 

330339  ;  42341 

90594 

57 

4 

627030 

4O  .  U 

957040 

.  O 

669991 

04.  y 

330009:  42367 

90582 

56 

5 

627300 

45.0 

/I  X  A 

956981 

9.8 

9Q 

670320 

54.8 
54  8 

329680 

I  42394 

90669 

55 

6 

627570 

4O  .  U 

956921 

.  O 

9q 

670649 

54  '  S 

329351 

1  42420 

90557 

54 

7 

627840 

44  q 

956862 

.  y 
9q 

670977 

329023 

42446 

90545 

53 

8 

628109 

4-t  .  y 

AA  O 

956803 

,  y 
9  9 

671306 

54  7 

328694 

42473 

90532 

52 

9 

628378  T"'« 

956744 

9*  9 

671634 

04  .  / 

54  7 

328366 

42499 

90520 

51 

10 

628647 

A  A  Q 

956684 

9Q 

671963 

328037 

42525 

90507 

60 

11 

9.628916 

44.  o 

44  7 

9.956625 

.  y 
9  9 

9.672291 

54.  7 
54  7 

10.327709 

42552 

90495 

49 

12 

629185 

44  .  / 

44  7 

956566 

99 

672619 

54  6 

327381 

|  42578 

90483 

48 

13 

629453 

4^r  .  1 

44  7 

956506 

99 

672947 

54  6 

327053 

42604 

90470 

47 

14 

629721 

44  a 

956447 

9  9 

673274 

54  « 

326726 

42631 

90458 

46 

15 

629989 

4:4  .  D 

44  6 

956387 

99 

673602 

O4  .  U 

54  6 

326398 

42657 

90446 

45 

16 

630257 

44  a 

956327 

.  ^ 
9  9 

673929 

54  5 

326071 

42683 

90433 

44 

17 

630524 

44  .  O 

44  6 

956268 

99 

674257 

54  6 

325743 

42709 

90421 

43 

18 

630792 

44  c 

956208 

10  0 

674584 

54  5 

325416 

42736 

90408 

42 

19 

631059 

44  ,  O 

956148 

10  0 

674910 

Ort  .  O 

325090 

42762 

90396 

41 

20 

631326 

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44  5 

956089 

10  0 

675237 

54  4 

324763 

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90383 

40 

21 

9.631593 

44.4 

9  .  956029 

10.0 

9-675564 

54*4 

10.324436 

42816 

90371 

39 

22 

631859 

44.4 

955969 

10.0 

675890 

54  '4 

324110 

42841 

90358 

38 

23 

632125 

44.4 

955909 

676216 

54  '3 

323784  j  42867 

90346 

37 

24 

25 

632392 
632658 

44  .'3 
44.3 

955849 
955789 

io!o 

10  0 

676543 
676869 

54^3 
54  3 

323457  42894 
323131  II  42920 

J0334 
90321 

36 
35 

26 

632923 

44  3 

955729 

10  0 

677194 

54  3 

322806  1|  42946 

9030y 

34 

27 

633189 

44  2 

955569 

10  0 

677520 

54*2 

322480  142972 

90296 

33 

28 

633454 

955609 

1j)  A 

677846 

322154  42999 

90284 

32 

29 

6337191  44*0 

955548 

i\J  .V 

i   ,   ;  | 

678171 

Mo 

3218291  143025 

90271 

31 

30 

633984  ll. 

955488 

1U.  0 

10  0 

678496 

.  -i 

54  2 

321504:  43051 

90259 

30 

31 

9.634249 

44  1 

9.955428 

10  1 

9-678821 

54  1 

10.  321179!!  43077 

90246 

29 

32 
33 

634514 
634778 

44!  0 
44  0 

955368 
955307 

10'  i 

679146 
679471 

«J4,  1 

54.1 
54  1 

320854  :  43  104  90233 
320529  '4b  130  90221 

28 
27 

34 

635042 

44  n 

955247 

10'  i 

679795 

54  1 

320205;  43156 

;>0208 

26 

35 

63530o 

44  ,  U 

43  9 

955186 

10  1 

680120 

04  .  j. 

319880  i  431S2 

90196 

25 

36 

635570 

43  <4 

955126 

680444 

54  o 

3195561(43209 

90183 

24 

37 

635834 

40  .  y 
43  9 

955065 

10  1 

680768 

O4  .  U 

Mo 

3  1  9232  l|  43235 

90171 

23 

38 

636097 

40  .  y 

43  8 

955005 

10  1 

681092 

.  V 

3189081(43261 

90158 

22 

39 
40 

636360 
636623 

43.8 

954944 
954883 

io!i 

681416 
681740 

53!g 

53  9 

318584  ||  43287 
318260  1143313 

90146 
90133 

21 
20 

41 

9.636886 

40  7 

9.954823 

10  1 

9.682063 

cq  q 

10.317937 

43340 

90120 

19 

42 

637148 

4o  .  / 

43.7 

954762 

10  1 

682387 

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53  9 

317613 

43366 

90108 

18 

43 

637411 

43.7 

954701 

io!i 

682710 

53  8 

317290 

43392 

90095 

17 

44 

637673 

43*7 

95-1040 

683033 

53  8 

316967 

43418 

90082 

16 

45 

637935 

43.6 

954579 

10*  1 

683356 

53  8 

316644 

43445 

90070 

15 

46 

638197 

40  c 

954518 

10*2 

683679 

CO   ft 

316321 

43471 

90057 

14 

47 

638458 

4o  .  O 

43  6 

954457 

10  2 

684001 

Oo  .  O 
xo  7 

315999 

43497 

90045 

13 

1  48 

638720 

43*5 

954396 

10  2 

684324 

OO  .  / 

5q  7 

315676 

43523 

90032 

12 

49 

638981 

43  5 

954335 

10  '2 

684646 

Oo  .  I 

xq  7 

315354 

43549 

90019 

11 

•  50 

639242 

43  5 

954274 

10  2 

684968 

Oo  .  / 

cq  7 

315032 

43575 

90007 

10 

51 

9.639503 

43  4 

9.954213 

10  2 

9  .  685290 

Oo  ,  i 
cq  ft 

10.314710 

43602 

89994 

9 

52 

639764 

4O  .  4 

43  4 

954152 

10*2 

685612 

Oo  .  O 
xq  f> 

314388 

43628 

89981 

8 

53 

640024 

43  4 

954090 

10  2 

685934 

Oo  .  O 
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314066 

43654 

89968 

7 

54 

640284 

43  3 

954029 

10  2 

686255 

Oo  .  O 

CO  iO 

313745 

43680 

89956 

6 

55 

640544 

40  q 

953968 

10  2 

686577 

Oo  .  D 
5q  5 

313423 

43706 

89943 

5 

56 

640804 

4o  .  O 

43.3 

953906 

10^2 

686898 

Oo  .  O 

53  5 

313102 

43733 

89930 

4 

57 

641064 

953845 

10  2 

687219 

KO  5 

312781 

43759 

89918 

3 

58 

641324 

• 

963783 

102 

687640 

oo  .  o 

KQ  K 

312460 

43785 

89905 

2 

59 

641584 

A  q  o 

953722 

1U  .  *> 

i  (i  q 

687861 

Oo  .  0 

CO   A 

312139 

43811 

89892 

1 

60 

641842 

4o  .  £ 

953660 

117.  O 

688182 

Oo  .  4 

311818 

43837 

89879 

0 

Cosine. 

Sine. 

Co  tang. 

Tang. 

N.  cos. 

N.siue. 

' 

64  Degrees. 

'JLAIUHII.     Log.  Sines  and  Tangents.  (26°)  Natural  Sines.          47 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.   N.  sine.jN.  cos. 

o 

9.641842 

9.953660 

9.688182 

10.311818  :  43837 

89879  !  GO 

1 

642101 

43.  1 

953599 

10.3 

688502 

53.4 

311498  ;  43863 

89867 

59 

2 
3 

642360 
642618 

43.1 
43.1 

953537 
953475 

10.3 
10.3 

688823 
689143 

53  .  4 
53.4 

311  177  j  43889 

310857  439  K) 

89854 
89841 

58 
57 

4 

642877 

43.0 

953413 

10.3 

689463 

53.3 

310537  43942 

89828 

56 

6 

643135 

43.0 

953352 

10.3 

689783 

53  .3 

KO  O 

310217  143968 

89816 

55 

6 

643393 

43.0 

953290 

10.3 

690103 

OO.  O 

309897 

43994 

89803 

54 

7 

643650 

43.0 

953228 

10.3 

690423 

53.3 

309577 

44020 

89790 

53 

8 

643908 

42.9 

953166 

10.3 

690742 

53.3 

KO  9 

309258 

44046 

89777  I  52 

9 
10 

644165 
644423 

42!9 

953104 
953042 

io!s 

691082 
691381 

Oo  .  Z 

53.2 

CO  O 

308938  1  44072 
308619  '  44098 

89764 
89752 

51 

50 

11 

9.644680 

42.8 

9.952980 

10.3 

9.691700 

Oo.  2 

CO 

10.308300  44124 

89739 

49 

12 

644936 

42.8 

ACt   Q 

952918 

10.4 

1  A   1 

692019 

Oo  . 

CO 

307981  1  44151 

89726 

48 

13 

645193 

42.0 

952855 

10.4 

692338 

Oo  , 

CO 

30  ?6b2i  44177 

89713 

47 

14 

645450 

42.7 

952793 

10.4 

692666 

OO.  1 

CO   1 

307344:  '44203 

89700 

46 

15 

645706 

42.7 

952731 

10.4 

692975 

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307025  44229 

89687 

45 

16 
17 

645962 
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42.7 
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693293 
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306707  44255 
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647494 

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52  .9 

304799 

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952231 

10.4 
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695518 

62.9 
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304482 

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296905 

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10.6 

703409 

52.  3 

296591 

45088 

89259 

12 

49 

654309 

41.7 

950586 

10.6 

703723 

52.  3 

296277 

45114 

89245 

11 

50 

654558 

41  .6 

950522 

10.6 

704036 

52.  3 

295964 

45140 

89232 

10 

61 

9.654808 

41.6 

9.950458 

10.7 

9.704350 

52  .2 

-o  o 

10-295650 

45166 

89219 

9 

52 

655058 

41.6 

950394 

10.7 

704663 

oz  .2 

295337 

45192 

89206 

8 

53 

54 

655307 
655556 

41.6 
41.5 

950330 
950366 

10.7 
10.7 

704977 
705290 

52  .  2 
52.2 

KO   Ct 

295023 
294710 

4521889193 
4524389180 

7 
6 

55 

655805 

41.5 

950202 

10.7 

705603 

oz  .  2 

294397 

45269 

89167 

6 

56 

656054 

41.5 

950138 

10.7 

705916 

52.  1 

294084 

45295 

89153 

4 

57 

656302 

41.4 

950074 

10.7 

706228 

52.1 

293772 

45321 

89140 

3 

58 
59 

650551 
656/99 

41  .4 
41.4 

950010 
949945 

10.7 
10.7 

706641 
706864 

52.  1 
52.1 

293459 
293146 

45347 
45373 

89127 
89114 

2 
1 

60 

657047 

41.3 

949881 

10.7 

707166 

52.1 

292834 

45399 

89101 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

V 

03  Degrees. 

43           Log.  Sines  and  Tangents.  (27°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10  ' 

Cosine. 

D.  10" 

Tang. 

1).  iU'f  (Jotang.  ||N.  siue. 

N.  cos. 

0 

9.657047 

41  *l 

9.949881 

107 

9.707166 

52  0 

10.292834 

45399 

89101 

60 

1 

657295 

41.o 
A  1  Q 

949816 

1U  .  / 
107 

707478 

xo  n 

292522 

45425 

89087 

59 

2 

657542 

41  .  o 
A1  9 

949752 

IU  .  i 

707790 

un£  .  U 

CO  0 

292210 

45451 

89074 

58 

B 

657790 

41  .  ^ 
41  ° 

94')688 

1ft  fi 

7J8102 

O^  .  U 

KO   0 

291898 

45477 

S9061 

57 

4 

658037 

A1  9 

949623 

1  A  W 

708414 

C  1   Q 

291586  i  45503 

S9048 

56  ! 

6 
6 

658284 
658531 

41  .  £ 

41.2 

A\   1 

949558 
949494 

4  U  .  O 

10.8 

1  A  Q 

708726 
709037 

01  .  y 

51.9 
ci  n 

291274 
290983 

]  45529 
45554 

89035 
89021 

55 
54 

7 

658778 

41  .  1 
41  1 

949429 

IU.  O 

108 

709349 

01  .  y 

EI  n 

290651 

45580 

89008 

53 

8 

659025 

41  .  1 
A  1  1 

949364 

IU  .  o 
108 

709660 

o  i  .  y 

Mn 

290340 

45606 

88995 

52 

9 

659271 

41  .  1 

A  1  n 

949300 

IU  .  o 
108 

70^971 

.  y 

fil  8 

290029 

45632 

88981 

51 

10 

659517 

41  .  U 

A  1  n 

949235 

IU  .  Q 

i  n  ft 

710282 

Ol  .  o 
ei  o 

289718 

45G58 

88968 

50 

11 

9.659/63 

41  .  U 

di  n 

9.949170 

IU.  o 
1  0  8 

9.710593 

Ol  .  o 

K  1   Q 

10.289407 

45684 

88955 

49 

12 

660  J09 

41  .  U 

949105 

IU.  o 

710904 

01  .  o 

289096 

45710 

88942 

48 

13 

660255 

40.9 

4-0  Q 

949040 

10.8 

108 

711215 

51  .8 

K1   Q 

288785 

45736 

88928 

47 

14 

660501 

948975 

iu.  o 

i  o  Q 

711525 

Ol  .  o 

ei  17 

288475 

45762 

88915 

46 

15 

600746 

A 

948910 

IU.  o 

711836 

0  1  .  / 

288164 

45787 

88902 

45 

16 

660991 

40.9 

948845 

10.8 

1  A  Q 

712146 

51  .7 

287854 

45813 

88888 

44 

17 

661236 

40.8 

Af\   0 

948780 

10.8 
i  A  n 

712456 

51.7 

287544 

45839 

88875 

43 

18 

661481 

4U.o 

948715 

10.  9 

712766 

51.7 

287234 

45865 

88862 

42 

19 

661726 

40.8 

948650 

10.9 

713076 

51  .6 

286924 

45891 

88848 

41 

20 

661970 

40.7 

u  i  ft 

948584 

10.9 

713386 

51  .6 

286614 

45917 

88835 

40 

21 

9.662214 

4U.7 

9.948519 

10.9 

9.713696 

51  .6 

10.286304 

45942 

88822 

39 

22 

662459 

A(\  "7 

948454 

10.9 

1  A  O 

714005 

51  .6 

K1  £; 

286996 

45968 

88808 

38 

23 

662703 

4U.  / 

At\  £1 

948388 

iu.y 

714314 

51  .6 

285686 

45994 

88795 

37 

24 

662946 

4U.D 
4ft  fl 

948323 

10.  9 

I  A  q 

714624 

c  1  t; 

285376 

46020 

88782 

36 

25 

663190 

4U  .  D 
AC\  (-* 

948257 

iu.y 

714933 

01.0 

285067 

46046 

88768 

35 

26 
27 

663433 
663677 

4U.O 

40.5 

4  A  K. 

948192 
948126 

10.  9 
10.9 

715242 
715551 

61  .5 
51.5 

284758 
28444.9 

46072 
46097 

88755 
88741 

34 
33 

28 

663920 

4U.O 

948060 

10.9 

715860 

51  .4 

284140 

46123 

88728 

32 

29 

664163 

40.5 

Af\   K 

947995 

10.9 
nf\ 

716168 

51  .4 

283832 

46149 

88715 

31 

30 
31 

664406 
9.664348 

4U.5 
40.4 

947929 
9.947863 

.0 

11.0 

716477 
9.716785 

51  .4 
51.4 

283523 
10.283215 

46175 
46201 

88701 
88688 

30 
29 

32 

664391 

40.4 

A(\  4 

947797 

11  .0 

nf\ 

717093  ef-J 

282907 

46226 

88674 

28 

33 

665:33 

4U  .  4 

947731 

.  u 

7174011°!^ 

282599 

46252 

88661 

27 

34 

665375 

40.3 

947665 

11.0 

717709!  9!  '„ 

282291 

46278 

88647 

26 

35 

665617 

40.3 

947600 

11.0 
no 

718017 

Ol  .0 
ri  o 

281983 

46304 

88634 

25 

36 

665359 

40  2 

947633 

.  V 

no 

718325 

01  .  o 

281675 

46330 

88620 

24 

37 

666100 

947467 

.  u 

nf\ 

718633 

61  .0 

281367 

46355 

88607 

23 

38 

666342 

A(\  9 

947401 

.  u 

718940 

51  .2 

281060 

46381 

88593 

22 

39 

666583 

o 

947335 

11.0 
U(\ 

719248 

51  .2 

280752 

46407 

88680 

21 

40 

666324 

A  n  1 

947269 

.U 

nf\ 

719555 

51.2 

280445 

46433 

88566 

20 

41 

9.667065 

4U.  1 

9.947203 

.0 

nf\ 

9.719862 

51  .2 

10.280138 

46458 

88553 

19 

42 

667305 

40.  1 

A  A  1 

947  \  36 

.u 

720169 

51  .2 

279831 

46484 

88539 

18 

43 
44 

667546 
667786 

4U.  1 

40.1 

947070 
947004 

11.1 

11.1 

720476 

720783 

51.1 
51.1 

279524 
279217 

46510 
46536 

88526 

88512 

17 
16 

45 

668027 

40.0 

4t\  f\ 

946937 

n  .1 
Hi 

721089 

51  .1 

278911 

46561 

88499 

15 

46 
47 

668-267 
668,  i06 

4U  U 

40.0 
oq  q 

946871 
946804 

.  i 
11.1 

721396 
721702 

51  .  1 

f!-i 

278604 
278298 

46587 
46613 

88485 

88472 

14 
13 

48 

668/46 

oy  .  y 
qq  q 

946738 

. 

722009  ";•" 

277991 

466C9 

88458 

12 

49 

668J86 

oy  .  y 
qq  q 

946671 

. 

722315  *}•« 

277685 

46664 

88445 

11 

50 

669,  !25 

oy  .  y 
qq  q 

946604 

» 

722621  g-g 

277379 

46690 

88431 

10 

51 

9.669464 

oy  .  y 

qq  o 

9.946538 

.  , 

9.722927  J}-£ 

10.277073 

46716 

88417 

9 

52 

<;<;:>  '0:5 

oy  .  o 

qq  x 

946471 

. 

723232  JJ'J 

276768 

46742 

88404 

8 

53 

6691*42 

oy  .  o 

946404 

723538  gj-g 

276462 

46767 

88390 

7 

54 

670181 

qq  7 

9.46337 

11. 

276156 

46793 

88377 

6 

55 

670119 

oy  .  / 

946270 

. 

724149  ijjJj-9 

275851 

46819 

88363 

5 

56 

670;558 

39.7 

946203 

11.2 

724454  |°"-jj 

276546 

46844 

88349 

4 

57 

670896 

J9.7 

946136 

11.2 

724769  !°X'X 

275241 

46870 

88336 

3 

58 

671134 

39.7 

946069 

11.2 

725066 

ou.o 

274936 

46896 

88322 

2 

59 

671372 

39.6 

946002 

11.2 

725369 

50.8 

274631 

46921 

88308 

1 

60 

671609 

39.6 

945935 

11.2 

725674 

60.8 

274326 

46947 

88295 

0 

dowine. 

Sine. 

Cotang. 

Tang. 

N.  cos 

N.sine. 

C2  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (28°)  Natural  Sines.          49 

' 

Sine. 

D.  10' 

Cosine. 

D.  10' 

Tang. 

D.10 

Cotang.  1  |N.  sine 

N.  cos 

0 

9.671609 

on  A 

9.945935 

Ho 

9.725674 

Krt  8 

10.274326 

46947 

88295 

60 

1 

671847 

OC7  .  O 

on  K 

945868 

.  * 

725979 

OU.C 

274021 

46973 

88281 

59 

2 

672034 

o9.  O 

94580!) 

11.2 

726284 

50.fi 

273716 

46999 

88267 

58 

3 

672321 

39.5 

945733 

11.2 

726588 

50.7 

273412 

47024 

88254 

57 

4 

672558 

39.  5 

945666 

11.2 

726892 

50.7 

273108 

47050 

88240 

56 

5 

672795 

39.  5 

945598 

11.2 

727197 

50.7 

272803 

47076 

88226 

55 

6 

673032 

39.  4 

945531 

11.2 

727501 

50.7 

272499 

47101 

88213 

54 

7 

673268 

39.  4 

945464 

11.2 

727805 

50-7 

272195 

47127 

88199 

53 

8 

673505 

39.  4 

945396 

11.3 

728109 

50-6 

271891  1  47153 

88185 

52 

9 

673741 

39.  4 

945328 

11  .3 

728412 

50-6 

271588  47178 

88172 

51 

10 
11 
12 

673977 
9.674213 
674448 

39.  3 
39.3 
39.3 

OQ  O 

945261 
9.945193 
945125 

11.3 
11.3 
11.3 

Uo 

728716 
9.72902C 
729323 

50-6 
50.6 
50.6 

KA  K 

271284 
10.270980 
270677 

47204 
47229 
47255 

88158 
88144 
88130 

50 
49 

48 

13 

674684 

oy  .  -« 

945058 

.  0 

729626 

50  -5 

270374 

47281 

88117 

47 

14 

674919 

39.  2 

OQ  O 

944990 

11.3 

no 

729929 

5Q.5 

270071 

47306 

88103 

46 

15 

675155 

oy  „  J 

OQ  0 

944922 

.  0 

no 

730233 

50-5 

FiA  K 

269767 

47332 

88089 

45 

16 

675390 

oy  .  & 

OQ  1 

944854 

.  o 
no 

730535 

O(J.O 

Kf\  ff 

269465 

47358 

88075 

44 

17 

675624 

oy  .  i 

OQ   -1 

944786 

•  o 

UQ 

730838 

DU.O 

269162 

47383 

88062 

43 

18 

675859 

oy  .  i 

OQ  1 

944718 

.0 

Uo 

731141 

50.4 

Kn  A 

268859 

47409 

88048 

42 

19 

676094 

O«J.  1 
OQ   I 

944650 

.  o 

731444 

O(J.4 

268556 

47434 

88034 

41 

20 

676328 

oy  .  i 

OQ   A 

944582 

11.3 

731746 

50.4 

268254 

47460 

88020 

40 

21 

9.676562 

oy  .  o 

OQ   A 

9.944514 

11.4 

9.732048 

50-4 

10.267952 

47486 

88006 

39 

22 

676796 

jy  .  (j 

OQ   A 

944446 

11.4 

732351 

50-4 

267649 

47511 

87993 

38 

23 

677030 

oy  .  u 

OQ   A 

944377 

11.4 

732653 

5Q.  3 

267347 

47537 

87979 

37 

24 

677264 

oJ  .  U 

OQ  Q 

944309 

11.4 

UA 

732955 

50-3 

267045 

47562 

87965 

36 

25 

677498 

oo  .  y 

OQ  Q 

944241 

.*± 
UA 

733257 

50-3 

266743 

47588 

87951 

35 

26 

677731 

oo  .  y 

OQ  Q 

944172 

.4 
UA 

733558 

50-3 

266442 

47614 

87937 

34 

27 

677964 

oo  »  y 

OQ  C 

944104 

.4 

UA 

733860 

50-3 

266140 

4763987923 

33 

28 

678197 

oo  .  O 

OQ  Q 

944036 

.1 

HA 

734162 

50-2 

265838 

4766587909 

32 

29 

678430 

oo  .  o 
oQ  G 

943967 

.4 

734463 

50-2 

265537 

4769087896 

31 

30 

678663 

oo  .  o 

OQ  Q 

943899 

11.4 

734764 

50-2 

265236 

4771687882 

30 

31 

9.678895 

OO  .  O 

OQ   7 

9.943830 

11.4 

9.735066 

50-2 

10.264934 

4774187868 

29 

32 

679128 

oc  »  t 

OQ  7 

943761 

11.4 

735367 

50.2 

264633 

4776787854 

28 

33 

679360 

OO.  I 

OQ  rf 

943693 

11.4 

735668 

50.2 

264332 

4779387840 

27 

34 
35 

679592 
679824 

oo.  / 

38.7 

oQ  {? 

943624 
943555 

11.5 
11.5 

735969 
736269 

50.1 
50.1 

264031 
263731 

47818,87826 
47844  87812 

26 
25 

36 

680056 

OO  .D 
OQ  p 

943486 

11.5 

736570 

50.1 

263430 

47869 

87798 

24 

37 

680288 

oo.o 
38  6 

943417 

11.5 

UK 

736871 

50.1 

S;A  i 

263129 

47895 

87784 

23 

38 

680519 

38*5 

943348 

•  o 

UK 

737171 

Du.  1 

^A  A 

262829 

47920 

87770 

22 

39 

680750 

38  5 

943279 

•  0 

UK 

737471 

OU.  U 
^A  n 

262529 

47946 

87756 

21 

40 

680982 

OQ  *  K 

943210 

.  O 

He; 

737771 

~>u.  u 

262229 

47971 

87743 

20 

41 

9.681213 

oo  .  O 

00   K 

9.943141 

.0 

9.738071 

50.0 

10.261929 

47997 

87729 

19 

42 

681443 

oo.  O 

00    A 

943072 

11.5 

738371 

50.0 

•*  A  A 

261629 

48022 

87715 

18 

43 

681674 

oo.  4 

00   ,. 

943003 

11.5 

738671 

oO.O 

261329 

48048 

87701 

17 

44 

681905 

00.4 

00    A 

942934 

11.5 

738971 

49.9 

261029 

48073 

87687 

16 

45 

682135 

00.4 

OQ   A 

942864 

11.5 

nc 

739271 

49.9 
4n  o 

260729 

48099 

B7673 

15 

46 

682365 

oo  .  4 

OO   0 

942795 

.  O 

U£? 

739570 

4y  .y 

4n  o 

260430 

4812487659 

14 

47 

682595 

oo  ,  o 

38  3 

942726 

.O 

Uc 

739870 

4y  .  y 

4.Q  Q 

260130 

48150 

B7645 

13 

48 

682825 

OQ'O 

942656 

.  o 
Ua 

740169 

*iy  .  y 

AQ  Q 

259831 

48175 

B7631 

12 

49 

683055 

oo  .  o 

OQ  0 

942587 

.  o 
Ua 

740468 

rty  .  y 

4Q  8 

259532 

48201 

37617 

11 

50 

683284 

OO  .  o 
OQ  9 

942517 

.  o 

nf! 

740767 

4y  .  o 
4.Q  R 

259233 

48226 

37603 

10 

51 

•.683514 

oo  ,  4 
OQ  9 

9.942448 

.  O 
Ua 

9.741066 

4y  .  o 

4.Q  8 

0.258934 

48252 

37589 

9 

52 

683743 

oo  .  4 

OQ  9 

942378 

.  o 

Uf; 

741365 

4y  .  o 
4Q  8 

258635 

48277 

37575 

8 

53 

683972 

oo  ,  ^ 

OQ  0 

942308 

.  D 
Uc 

741664 

iy  .o 

4Q  8 

258336 

48303 

37561 

7 

54 

684201 

OO  .  w 
00  1 

942239 

.O 
Uc 

741962 

4y  ,  o 

4Q  7 

258038 

48328 

37546 

6 

55 

684430 

OO  .  1 
OQ  1 

942169 

.O 
nf* 

742261 

•iy  .  / 

257739 

48354 

37532 

5 

56 

684658 

OO.  1 
OQ  1 

942099 

»D 
no 

742559 

49.7 

257441 

48379 

37518 

4 

57 

684887 

OO  .  1 

942029 

.0 
U/> 

742858 

49.7 

257142 

48405 

37504 

3 

58 

685115 

38.0 

941959 

.0 

743156 

49.  7 

256844 

48430 

37490 

2 

59 

685343 

38.  0 

941889 

11.6 

743454 

49.  7 

256546 

48456 

37476 

1 

GO 

685571 

38.0 

941819 

11.7 

743752 

49.7 

256248 

48481 

37462 

0 

Cosine. 

Sine. 

Cotaiig. 

Tang. 

N.  cos. 

N.sine. 

/ 

61  Degrees. 

50          Log.  Sines  and  Tangents.  (29°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10" 

Cosine.  |D.  10" 

Tang. 

D.  10" 

Cotang. 

N.  sine. 

N.  cos. 

0 

9.685571 

9.941819 

9.743752 

0.256248 

48481 

87462 

60 

1 

685799 

38.0 

941749 

11  .7 

744050 

or 

255950 

48506 

37448 

59 

2 

686027 

37.9 

941679 

11  .7 

nrt 

744348 

or 

255652 

48532 

37434 

68 

3 

686254 

QT  f\ 

941609 

.  4 

UT 

744645 

AQ  f\ 

255355 

48557 

37420 

67 

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9 

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941187 

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746429 

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746726 

49  5 

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48735 

87321 

50 

11 

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9.941046 

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no 

9.747023 

49  4 

10.252977 

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87306 

49 

12 

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no 

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49  '3 

250903 

48938 

87207 

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19 
20 

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o  /  ,  O 

37.5 

940480 
940409 

,  o 
11.8 

no 

749393 

749689 

49*3 

250607 
250311 

48964 
48989 

87193 

87178 

41 

40 

21 
22 

3.690323 
690548 

37.5 
37.4 

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9.940338 
940267 

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11.8 

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9.749985 
750281 

49*3 

AQ  0 

10.250015 
249719 

4901487164 
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38 

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o/  .4 

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940196 

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no 

750576 

*±*y  .  *. 
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ol  .4 

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940054 

11.9 
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248833 

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87107 

35 

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691444 

o  I  .  o 

OT  O 

939982 

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no 

761462 

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248538 

49141 

87093 

34 

27 

691668 

o7  .0 

939911 

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751757 

rit/  .  M 

4Q  9 

248243 

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87079 

33 

28 

691892 

37.3 

OT  O 

939840 

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Un 

752052 

T:«J  .  ^ 
49   1 

247948 

49192 

87064 

32 

29 

692115 

o7  .0 

939768 

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nn 

752347 

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247653 

49217 

87050 

31 

30 
31 

692339 
9.692562 

37  ..2 
37.2 
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939697 
9.939625 

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11.9 
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752642 
9.752937 

T:*-'  ,  1 

49.1 
49  1 

247358 
10.247063 

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49268 

87036 
87021 

30 
29 

32 

692785 

o  1  .  6 

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939554 

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753231 

49*1 

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33 

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246474 

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693231 

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939410 

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753820 

4*7  .  1 

49  ft 

246180 

49344 

86978 

26 

35 

693453 

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OT  1 

939339 

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Uq 

754115 

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49  ft 

245886 

49369 

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25 

36 

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939267 

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12  0 

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T:*7  ,  U 

49  0 

246591 

49394 

86949 

24 

37 

693898 

37.0 

OT  A. 

939195 

754703 

49  ft 

245297 

49419 

86935 

23 

38 

694120 

o7  .  U 

OT  A 

939123 

|0*A 

754997 

*±*7  .  U 
4Q  A 

245003 

49445 

86921 

22 

39 

694342 

61  .  U 

939052 

1  Q  A 

755291 

T:*7  .  U 

49  ft 

244709 

49470 

86906 

21 

40 

694564 

37.0 

oft  O 

938980 

U.  U 
1  Q  f\ 

755585 

1*7  .  U 

48  9 

244415 

49495 

86892 

20 

41 
42 

9.694786 
695007 

06.  y 

36.9 
36  9 

9.938908 
938836 

I  .w  .  U 

12.0 
12  ft 

9.755878 
756172 

48  .'9 
48  9 

10.244122 
S43828 

49521 
49546 

86878 
86863 

19 
18 

43 

695229 

938763 

1*  .  U 
1  2  ft 

756465 

4g'g 

243536 

49571 

86849 

17 

44 

695450 

oft  ft 

938691 

!•*$  .  U 

12  0 

756759 

48*9 

243241 

49596 

86834 

16 

45 

695671 

OO.O 

Oft  Q 

938619 

12  ft 

757052 

48*9 

242948 

49622 

86820 

15 

46 

695892 

OO.O 
oft  Q 

938547 

JU*  .  U 
19  ft 

757345 

242655 

49647 

86805 

14 

47 

690113 

OO.O 

938475 

1>£  .  U 

12  0 

757638 

48*8 

242362 

49672 

86791 

13 

48 

696334 

36.8 

oft  T 

938402 

19  1 

757931 

^±o  .0 

242069 

49697 

86777 

12 

49 

696554 

ot>.  / 
oft  7 

938330 

i  ^  .  i 
12  1 

758224 

48.8 

241776 

49723 

86762 

11 

50 

696775 

oO  .  1 

938258 

12*  1 

758517 

48  8 

241483 

49748 

86748 

10 

51 

9.696995 

36.  7 

9.938185 

9.758810 

40  .  o 
48  8 

10.241190 

49773 

86733 

9 

52 

697215 

36.  7 

938113 

12  1 

759102 

4o  .  o 

48  7 

240898 

49798 

86719 

8 

53 

697435 

36.6 

Oft  ft 

938040 

12  1 

769395 

4o  .  I 

48  7 

240605 

49824 

86704 

7 

54 

697654 

OO.  O 

937967 

1^  .  L 

759687 

48  7 

240313 

49849J86690 

6 

55 

01)7874 

36.6 

937895 

19*1 

759979 

4o  ,  / 

48  7 

240021 

4987486675 

6 

56 

698094 

36.6 

Oft  K 

937822 

1-i.  1 
19  1 

760272 

4o  .  * 

48.7 

239728 

4989986661 

4 

67 

698313 

oO.  o 

937749 

AZ  ,  1 
-JO  1 

760564 

48*7 

239436 

4992486646 

3 

58 

098532 

36.5 
oft  K 

937676 

760856 

4o  .  i 

48  6 

239144 

4995086632 

2 

59 

698751 

OO.O 

Oft  K 

937604 

10  1 

761148 

48  6 

238852 

49975 

80617 

1 

60 

698970 

oO.O 

937531 

1  -  .  1 

761439 

238561 

5000U 

80603 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  co* 

N.pim- 

' 

60  Degrees. 

TABLE  11.     Log.  Sines  and  Tangents.  (30°)  Natural  Sines.          51 

' 

Sine. 

D.  1U 

Oft  A 

Cosine. 

D.  10 

Tang. 

D.  10 

Co  tang. 
10.238561 

N.  sint 

N.  cos 

60 

0 

9.698970 

9.937531 

19  1 

9.761439 

AQ  f! 

5000 

86603 

1 

699189 

OO  •  T 

or*  A 

937458 

1.0.  J 

1  O 

761731 

4o.O 

238269 

5002 

86588 

59 

c 

699407 

OO.  4 
qc  A 

937385 

15*. 

19 

762023 

48.6 

AQ  fc 

237977 

5005 

86573 

58 

£ 

699626 

OO  .  T 

Oft  A 

937312 

IA  , 
1  9 

762314 

43.  0 
AQ  ft 

237686 

50076 

86559 

57 

^ 

699844 

OO  .  T 

°lfi  ' 

937238 

1^  . 
1  9 

762603 

4o  .0 

AQ   £ 

237394 

5010 

86544 

56 

£ 

700062 

oo  .  * 

O£ 

937165 

iA  . 

762897 

4o  .O 

237103 

50126 

86530 

55 

6 

700280 

oo. 

Of?   • 

937092 

12.  ' 
10  ' 

763188 

48.5 

AQ  r^ 

236812 

5015 

86515 

54 

7 

700498 

oo  .  < 

oc* 

937019 

!•*« 

763479 

4o.O 

236521 

50176 

86501 

53 

8 

700716 

oo. 

0£»   «• 

936946 

12.5 

763770 

48.  1 

236230 

5020 

86486 

52 

700933 

oo. 

Oft 

936872 

12. 

1  9 

764061 

48.5 

AQ  K. 

235939 

5022~ 

86471 

51 

10 

701151 

oo  . 

Oft 

936799 

\.£  , 

19 

764352 

4o  .  O 

AQ  / 

235648 

50252 

86457 

50 

11 

9.701368 

OO  . 
or* 

9.936725 

4«w« 

1  O 

9.764643 

4o  .< 

10.235357 

50277 

86442 

49 

12 

701585 

oo. 
or1  c 

936652 

\2i  . 

764933 

48.^ 

235067 

50302 

86427 

48 

13 

701802 

oo. 

oo  1 

936578 

12.3 

765224 

48.^ 

234776 

5032~ 

86413 

47 

14 

702019 

OO.  1 

or*  1 

936505 

12.3 

765514 

48.4 

234486 

50352 

86398 

46 

15 

702236 

OO.  I 
O^J  1 

936431 

12.3 

765805 

48.4 

234195 

5037" 

86384 

45 

16 

702452 

Jo.  i 

or*  1 

936357 

12.3 

766095 

48.4 

233905 

50403 

86369 

44 

17 

702669 

OO.  1 

936284 

12.3 

766385 

48.4 

233615 

50428 

86354 

43 

18 

702885 

36.  0 

qo  A 

936210 

12.3 

10  o 

766675 

48.3 

AQ   0 

233325 

50453 

86340 

42 

19 

703101 

OO  •  v/ 

936136 

116.  O 

766965 

4o.d 

233035 

50478 

86325 

41 

20 

703317 

36.0 
Q£J  ft 

936062 

12.3 

10  o 

767255 

48.3 

232745 

50503 

86310 

40 

21 

9.703533 

oo.U 

9.935988 

lie.  a 

9.767545 

48.3 

10.232455 

50528 

86295 

39 

22 
23 

703749 
703964 

35.9 
35.9 

QK  Q 

935914 
935840 

12.3 
12.3 

10  o 

767834 
768124 

48.3 
48.3 

232166 
231876 

50553 
50578 

86281 
86266 

38 
37 

24 

704179 

oo.y 

q~  o 

935766 

1-6.  o 

1  r\   A 

768413 

48.2 

231587 

50603 

86251 

36 

25 

704395 

oo.y 

QK  Q 

935692 

ll«.4 

1O   xi 

768703 

48.2 

231297 

50628 

86237 

35 

26 

704610 

oo.y 

q-  0 

935618 

lii.4 
10  1 

768992 

48.2 

231008 

50654 

86222 

34 

27 

704825 

OO.O 

q~   Q 

935543 

lxi.4 
in  /i 

769281 

48.2 

230719 

50679 

86207 

33 

28 

705040 

OO  .0 

q-   Q 

935469 

12.4 

1O   yl 

769570 

48.2 

230430 

50704 

86192 

32 

29 

705254 

oo.o 

q-   Q 

935395 

lz.4 

769860 

48.2 

230140 

50729 

86178 

31 

:;* 

705469 

OO.  0 

O-  T 

935320 

12.4 

770148 

48.  1 

229852 

50754 

86163 

30 

31 

9.705683 

00.  / 

O"  n 

9.935246 

12.4 

9.770437 

48.1 

10-229563 

50779 

86148 

29 

32 

705898 

oo.  i 

O-1  1 

935171 

12.4 

770726 

48.1 

229274 

50804 

86133 

28 

33 

706112 

01.  I 

O~  1 

935097 

12.4 

771015 

48.1 

228985 

50829 

86119 

27 

34 

706326 

00.  1 

93502-2 

12.4 

771303 

48.1 

228697 

50854 

86104 

26 

35 

706539 

35.6 

934948 

12.4 

771592 

48.1 

228408 

50879 

86089 

25 

36 

706753 

35.6 

934873 

12.4 

771880 

48.1 

228120 

50904 

86074 

24 

37 

706967 

35.6 

934798 

L2.4 

772168 

48.0 

227832 

50929 

86059 

23 

38 

707180 

35.6 

934723 

L2.5 

772457 

48.0 

227543 

50954 

86045 

22 

39 

707393 

35.5 

Q"   K 

934649 

12.5 

772745 

48.0 

227255 

50979 

86030 

21 

40 

707606 

OO.O 

O"  K 

934574 

12.  5 

773033 

48.0 

226967 

51004 

86015 

20 

41 

9.707819 

OO.O 

OK  e 

9.934499 

12.5 

9.773321 

48.0 

10-226679 

51029 

86000 

19 

42 

708032 

OO.O 

OK  A 

934424 

L2.5 

773608 

48.0 

226392 

51054 

85985 

18 

43 

708245 

OO.4 
O"   /I 

934349 

L2.5 

773896 

47.9 

226104 

51079 

85970 

17 

44 

708458 

oo.4 

934274 

12.5 

774184 

47.9 

225816 

51104 

85956 

16 

45 

7036701 

35.4 

934199 

12.  5 

774471 

47.9 

225529 

51129 

5941 

iO 

46 

708882 

35.4 

934123 

12.  5 

774759 

47.9 

225241 

51154 

5926 

14 

47 

709094 

35.3 

OX  Q 

934048 

12.5 

1O  K 

775046 

47.9 

A^l  O 

224954 

51179 

5911 

13 

48 

70930ci 

OO.O 

933973 

Lii.O 

775333 

47.  y 

224667 

51204 

5896 

12 

49 

709518 

35.3 

933898 

12.5 

775621 

47.9 

224379 

51229 

5881 

11 

50 

709730 

35.3 

O"  O 

933822 

12.6 

775908 

47.8 

4*y  Q 

224092 

51254 

5866 

10 

51 

)  .  709941 

OO.O 

9.933747 

L2.6 

9.776195 

7.0 

0-223805 

51279 

5851 

9 

52 

710153 

35.2 

933671 

L2.6 

776482 

47.8 

223518 

51304 

5836 

8 

53 

710364 

35.2 

933596 

L2.6 

776769 

47.8 

223231 

51329 

5821 

7 

54 

710575 

35.2 

933520 

L2.6 

777055 

47.8 

222945 

51354 

5806 

6 

55 

71U786 

35.2 

933445 

[2.6 

777342 

47.8 

222658 

51379 

5792 

5 

56 

710967 

35.1 

933369 

i2.6 

777628 

47.8 

222372 

51404 

5777 

4 

57 

711208 

35.  1 

933293 

[2.6 

777915 

47.7 

222085 

51429 

5762 

3 

58 

71M19 

35.  I 

933217 

^2.6 

778201 

47.7 

221799 

51454 

5747 

2 

59 

711629 

oo.  1 

933141 

.2.6 

778487 

47.7 

221612 

51479 

5732 

1 

60 

711839 

35.0 

933066 

.2.6 

778774 

17.7 

221226 

"1504 

5717 

0 

Cosine. 

Sine. 

Cotang;. 

Tang. 

N.  cos. 

\.Kine. 

' 

59  Degrees. 

52          Log.  Sines  and  Tangents.  (31°)  Natural  Sines.     TABLE  II. 

' 

Sine.   |D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang. 

N.sine.jN.  cos. 

0 

9.711839 

9.933036 

9.778774 

10.221226 

61504185717 

60 

1 

712050 

35.0 

qx  A 

932990 

12.6 

779060 

47.7 
47  7 

220940 

5162985702 

59 

2 

712260 

oO  .  U 

932914 

I  ~  .  / 

779346 

220654 

5  1554  185687 

58 

3 

712469 

35.0 

932838 

12.7 

779632 

47.6 

220368 

51579185672 

57 

4 

712679 

34.9 

932762 

12.7 

779918 

47  .6 

220082 

61604185657 

56 

6 

712889 

34.9 

932685 

12.7 

780203 

47.6 

219797 

61628185642 

55 

6 

713098 

34.9 

932609 

12.7 

780489 

47.6 

219511 

51653 

85627 

54 

713308 

34.9 

932533 

12.7 

780775 

47.6 

219225 

51678 

85612 

53 

8 

713517 

34.9 

932457 

12.7 

781060 

47  .6 

218940 

61703 

85597 

52 

9 
10 

713726 
713935 

34.8 
34.8 

932380 
932304 

12.7 
12.7 

781346 
781631 

47.6 
47.5 

218654 
218369 

51728 
51753 

85582 
85567 

51 
50 

11 

9.714144 

34.8 

9.932228 

}%'l  19.  781916 

47.5 

10.218084 

51778 

85551 

49 

12 
13 
14 

714352 
714561 

714769 

34.8 
34.7 
34.7 

932151 
932075 
931998 

li*  .  7 
12.7 
12.8 

782501 

782486 
782771 

47.5 
47.5 
47.5 

217799 
217514 
217229 

51803  J85636 
51828i8552l 
51852J85506 

48 
47 
46 

15 

714978 

34.7 

931921 

12.8 

783056 

47.6 

216944 

51877185491 

45 

16 

715186 

34.7 

°.d  7 

931845 

12.8 

19  S 

783341 

47.5 
47  5 

216659 

5190285476 

44 

17 

18 

715394 
715602 

o4  .  i 

34.6 

931768 
931691 

1-4  .  O 

12.8 

783626 
783910 

47^4 

216374 
216090 

61927185461 

51952185446 

43 

42 

19 

715809 

34.6 

931614 

12.8 

784195 

47.4 

Art  A 

215805 

51977J85431 

41 

20   716017 

34.6 

Q/1  fi 

931537 

12.8 

784479 

47  .4 

4T  A 

215521 

52002 

85416 

to 

21  9.716224 

o4.D 

9.931460 

12.  8 

9.784764 

4  /  .  4 

10.215236 

52026 

85401 

39 

22   716432 

34.5 

931383 

12.8 

785048 

47.4 

214952 

52051  185385 

38 

23   716639 

34.5 

931306 

12.8 

785332 

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214668 

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24   716846 

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MR 

931229 

12.8 

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785616 

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214384 

52101 

85355 

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25 

26 

717053 

717259 

.  O 

34.5 

0  *   A 

931152 
931075 

1.4.  y 
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785900 
786184 

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214100 
213816 

52126 
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34 

27 

717466 

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930998 

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213632 

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930921 

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930843 

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212964 

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718085 
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930766 
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12.9 
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787319 
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212681 
10.212397 
212114 

52250 
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85234 

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29 
28 

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930533 

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211830 

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34 

718909 

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930456 

1.2.9 

788453 

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211547 

62349 

85203 

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35 

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930378 

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719320 

34.2 

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210981 

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37 

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930223 

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789302 

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210698 

52423 

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719730 

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930145 

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929989 

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207590 

52696 

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929286 

13  . 

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792692 

47.0 

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207308 

52720 

84974 

11 

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929207 

lo  . 

792974 

4/  .U 

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207026 

62745 

84959 

10 

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9.929129 

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52794 

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52869 

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13. 

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52893 

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57 

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13. 

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46.9 

205055 

52918 

84851 

3 

58 

723805  "£-L 

928578 

lo. 

1  O 

795227 

46  .  9 

204773 

52943 

84836 

2 

59 

724007  ]  XX  ',1 

928499 

lo. 

1  O 

795508 

46  .  9 

1f  Q 

204492 

5296'i 

84820 

1 

60 

724210, 

928420 

lo. 

795789 

46.  o 

204211  |i  62992 

84805 

0 

"Cosine.  I 

Sine. 

Cotang. 

Tang.   'lN.coB.JN.sine. 

T~ 

58  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (32°)  Natural  Sines. 

53 

' 

Sine. 

D.  10" 

Cosine.  |D.  10" 

Tang.  (D.  10"j  Cotaiig. 

N.  sine. 

N  .  cos  . 

0 

9.724210 

9.928420 

9.795789 

AC  Q 

10.204211 

52992 

84805 

60 

1 

724412 

33.7 

OO  O' 

928342 

13.2 
100 

796070 

46.h 

Aft.  ti 

203930 

53017 

84789 

59 

2 

724614 

OO  .  / 

928263 

lo  .  Z 

796351 

4u  .0 

203649 

53041 

84774 

58 

3 

724816 

33.6 

928183 

13.2 

796632 

46.8 

203368 

53066 

84759 

57 

4 

725017 

33.6 

928104 

13.2 

796913 

46.8 

203087 

53091 

84743 

56 

5 

725219 

33.6 

928025 

13.2 

797194 

46.8 

202806 

53115 

84728 

55 

6 

725420 

33.6 

927946 

13.2 

797475 

46.8 

2u2525 

63140J84712 

54 

7 
8 

725622 
725823 

33.5 
33.5 

927867 
927787 

13.2 
13.2 

797755 
798036 

46.8 
46.8 

202245 
201964 

53164J84697 

5318984681 

53 
52 

9 

726024 

33.5 

927708 

13.2 

798316 

46.7 

201684 

5321484666 

51 

10 

726225 

33.5 

927629 

13.2 

798596 

46.7 

201404 

5323884650 

50 

11 

9.726426 

33.5 

9.927549 

13.2 

9.798877 

46.7 

4n  rj 

10.201123 

53263|84635 

49 

12 
13 

726626 

726827 

33.4 
33.4 

927470 
927390 

13.2 
13.3 

799157 
799437 

4o.  / 
46.7 

.*-*  n 

200843 
200563 

53288 
53312 

84619 
84604 

48 

47 

14 

727027 

33.4 

927310 

13  .  3 

799717 

46  .  7 

AC   17 

200283 

53337 

84588 

46 

15 

727228 

33.4 

927231 

13.3 

799997 

4t>.  7 

200003 

53361 

84573 

45 

16 

727428 

33.4 

OO  Q 

927151 

13.3 

1  Q  O 

800277 

46.6 

AC  « 

199723 

53386 

84557 

44 

17 

727628 

06  .  o 

927071 

lo.  O 

1  Q  o 

800557 

4b  .  D 

Ad  fi 

199443 

53411 

84542 

43 

18 

727828 

33  .  3 

OO  Q 

926991 

10.  O 
1  0   q 

800836 

4o  .  o 
4f>  fi 

199164 

53435 

84526 

42 

19 

728027 

OO  .  O 
OQ  9. 

92691  1 

lo  .  o 
1  q  q 

801116 

4O  .  D 
Afi  fi 

198884 

53460 

84511 

41 

20 
21 
22 
23 

728227 
9.728427 
728626 
728825 

00  .  0 

33.3 
33.2 
33.2 

926831 
9.926751 
926671 
926591 

lo.  d 
13.3 

13.3 
13.3 

801396 
9.801675 
801955 

802234 

4O  .  O 

46.6 
46.6 
46.6 

198604 
10.198325 
198045 
197766 

53484  84495 
5350984480 
5353484464 
53558  84448 

40 
39 
38 
37 

24 

729024 

33.2 

926511 

13.3 

802513 

46.5 

197487 

53583 

84433 

36 

25 

26 

729223 
729422 

33.2 
33.1 

926431 
926351 

13.4 
13.4 

802792 
803072 

46.5 
46.6 

197208 
196928 

5360784417 
5363284402 

35 
34 

27 

28 

729621 

729820 

33.1 
33.1 

926270 
926190 

13.4 
13.4 

803351 
803630 

46.5 
46.5 

196649 
196370 

5365684386 
5368184370 

33 
32 

29 

730018 

33.  1 
OQ  n 

926110 

13.4 

i  q  A 

803908 

46.5 

AC  r. 

196092 

53705J84355 

31 

:;o 

31 

730216 
9.730415 

OO.  U 

33.0 

926029 
9.925949 

lo.4 

13.4 

804187 
9.804466 

40  .  0 

46.5 

195813 
10.195534 

53730J84339 

53754184324 

30 

29 

32 

730613 

33.0 
oo  n 

925868 

13.4 

804745 

46. 

195255 

53779^84308 

28 

33 

730811 

33.0 

925788 

13.4 

805023 

46. 

194977 

53804^4292 

27 

34 

731009 

33.0 

925707 

13.4 

805302 

46. 

194698 

63828  84277 

26 

35 

731206 

32.9 

925626 

13.4 

805580 

46. 

194420 

63853 

84261 

25 

36 
37 

731404 
731602 

32.9 
32.9 

Of)  Q 

925545 
925465 

13.4 
13.5 

i  q  e 

805859 
806137 

46. 
46. 
/in  4 

194141 

193863 

53877  [84245 
53902184230 

24 
23 

38 

731799 

O-i  .  i> 

925384 

lo  ,  D 

806415 

40  .  ^ 

193585 

53926 

84214 

22 

39 

731996 

32.9 

on  W 

925303 

13.5 

806693 

46.3 

AC   O 

193307 

53951 

84198 

21 

40 

732193 

32.0 

or»  Q 

925222 

13  .  5 

806971 

46  .0 

AC  Q 

193029 

53975 

84182 

20 

41 

9.732390 

d'2.o 

on  Q 

9.925141 

13.  5 

9.807249 

46.o 

10.192751 

54000184167 

19 

42 

732587 

w.o 

O'-x   0 

925060 

13.5 

807527 

46.3 

192473 

5402484151 

18 

43 
44 

732784 
732980 

32.0 

32.8 

or»  i°1 

924979 
924897 

13.5 
13.5 

807805 
808083 

46.3 
46.3 

192195 
191917 

5404984135 
5407384120 

17 
16 

45 

733177 

32.7 

924816 

13.5 

808361 

46.3 

191639 

54097  84104 

15 

46 

733373 

32.7 

924735 

13.5 

1O  C. 

808638 

46.3 

AC  a 

191362 

54122  84088 

14 

47 

733569 

32.7 

924654 

lo.b 

808916 

46.2 

191084 

54146  84072 

13 

48 

733765 

32.7 

on  O' 

924572 

13.6 

809193 

46.2 

190807 

54171  84057 

12 

49 
50 

733961 
734157 

32.  / 

32.6 

924491 
924409 

13.6 
13.6 

809471 
809748 

46.2 
46.2 

190529 
190252 

5419584041 
5422084025 

11 
10 

51 

9.734353 

32.6 

9.924328 

13.6 

9.810025 

46.2 

10.189975 

54244  !84009 

9 

52 

734549 

32.6 

924246 

13.6 

810302 

46.2 

189698 

54269  !83994 

8 

53 

734744 

32.6 

924164 

13.6 

810580 

46.2 

189420  54293 

83978 

7 

54 

734939 

32.5 

924083 

13.6 

810857 

46.2 

189143 

54317 

83962 

6 

55 
56 

735135 
735330 

32.5 
32.5 

924001 
923919 

13.6 
13.6 

811134 
811410 

46.2 
46.1 

188866 
188590 

5434283946 
54366:83930 

5 

4 

57 

735525 

32.  5 

923837 

13.6 

811687 

46.  1 

188313 

54391  83915 

3 

58 

735719 

32.  5 

923755 

13.6 

811984 

46.  1 

188036 

54415  83899 

2 

59 

735914 

32.4 

923673 

13.7 

812241 

46.  1 

187759 

54440  83883 

1 

60 

736109 

32.4 

923591 

13.7 

812517 

46.1 

187483 

54464 

83867 

0 

""Cosine. 

Sine. 

Cotang. 

Tang. 

NTcos. 

N.  si  ne. 

~i~~ 

57  Degrees. 

54          Log.  Sines  and  Tangents.  (33°)  Natural  Sines.     TABLE  IT. 

' 

Sine. 

IX  10 

Cosine. 

I).  10 

Tang. 

D.  10 

Cotang. 

,N.  sine. 

N.  cos 

0 

9.736109 

OO   A 

9.923591 

9.812517 

10.187482 

54464 

83867 

60 

1 

736303 

62  .  < 

923509 

13.1 

812794 

46.] 

187206 

64488 

83851 

59 

2 

736498 

32.^ 

923427 

13.7 

i  q  7 

813070 

46.] 
46  .  ] 

186930 

64513 

83835 

58 

c 

736692 

62  . 

923345 

lo  .  i 

813347 

186653 

54537 

83819 

57 

4 

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32.  i 

qo  « 

923263 

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1  q  7 

813623 

46.  C 

186377 

54561 

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56 

5 

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1  q  7 

813899 

AT  ( 

186101 

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55 

f> 

737274 

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lo.  / 

814175 

4o  .(. 

185825 

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54 

7 

737467 

32.  J 

923016 

13.7 

814452 

46.  ( 

185548 

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737661 

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814728 

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185272 

54659 

83740 

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737855 

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922851 

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815004 

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184996 

54683 

83724 

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738048 

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922768 

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815279 

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184721 

54708 

83708 

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9.738241 

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819684 

40.0 

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180316 

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741316 

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180041 

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5516983405 

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5519483389 

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31 
32 
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9.742080 

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31.8 
31.8 
31.8 
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9.921023 
920939 
920856 
920772 

13.9 
13.9 
14.0 
14.0 

9.821057 
821332 
821606 
821880 

45.7 
45.7 
45.7 
45.7 

10.178943 
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178394 
178120 

55218  83373 
55-242  '83356 
55266183340 
55291  !83324 

29 
28 
27 
26 

35 
36 

742842 
743033 

31  .7 
31.7 

Q1  *7 

920688 
920604 

14.0 

14.0 
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822154 

822429 

45.7 
45.7 

177846 
177571 

55315!83308 
55339  83292 

25 
24 

37 

743223 

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45.7 

177297 

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177023 

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9.745871 

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9.919339 

4. 

.826532 

45.5 

0.173468 

55702  83050 

9 

52 

746059 

919254 

4. 

826805 

1:5.5 

173195 

55726  83034 

8 

53 

54 

746248 
746436 

.3 

919169 
919085 

4.1 
4.1 

827078 
827351 

15  .'6 

5K 

172922 
172649 

5575083017 
5577583001 

7 
6 

55 
56 
57 

746624 

746812 
746999 

.3 
.3 
3 

919000 
918915 
918830 

4.  1 
4.1 

4.2 

827624 
827897 
828170 

.  o 
45.5 
45.4 

fL  A 

172376 
172103 
171830 

6579982985 
5582382969 
5584782953 

5 
4 
3 

58 

747187 

2 

918745 

40 

828442 

tO  ,  4 

171558 

6587182936 

2 

59 

747374 

o   o 

918659 

•  2 

828715 

rO.4 

171285 

5589582920 

1 

60 

747562 

o  .2 

918574 

4.2 

828987 

:5.4 

171013 

65919|82904 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

\.  cos.  JN.  sine. 

' 

56  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (34°)  Natural  Sines.          55 

' 

Sine. 

D.  10' 

Cosine. 

D.  10' 

Tang. 

D.  10 

Cotang.  |  N.sine 

N.  cos 

0 

9.747562 

qi  9 

9.918574 

Ho 

9.828987 

AK.   A 

10.171013 

55919 

82904 

60 

1 

747749 

OJL  ••« 

01  o 

918489 

.  -w 
1  4  O 

829260 

4O  .  4 
Af  A 

170740 

55943 

82887 

59 

2 

747936 

ol  .  £ 

qi  o 

918404 

14  .  Z 

829532 

4O  .  4 

A?\  A 

170468  j  55968 

82871 

58 

3 

748123 

918318 

*• 

829805 

4O  .  4 

170195  155992 

82855 

57 

4 

748310 

31  .1 

918233 

14.2 

830077 

45.4 

169923 

56016 

82839 

56 

5 

748497 

31  .1 

918147 

14.2 

1  1  O 

830349 

45.4 

A  ~  O 

169651 

56040 

82822 

55 

6 

748683 

31  .  1 

918062 

14.  Z 
Ho 

830621 

4o.d 

A  ~  O 

169379 

56064 

8280* 

54 

7 

748870 

31  .  1 

917976 

.  A 

830893 

4o  .0 

169107 

56088 

82790 

53 

8 

749056 

31.1 

0  t    (  \ 

917891 

14.3 

14  q 

831165 

45.3 

168835 

56112 

82773 

52 

9 

749243 

ol  .  U 

917805 

14  .  o 

831437 

4o  .  <J 

168563 

56136 

82757 

51 

10 

749426 

31  .0 

ol   A 

917719 

14.3 

Uq 

831709 

45.3 

AZ.  o 

168291 

56160 

82741 

50 

11 

12 

9.749615 
749801 

ol  .  U 

31.0 

9.917634 
917548 

.  o 
14.3 

Ho 

9.831981 
832253 

4O  .  d 

45.3 

10.168019 
167747 

56184 
56208 

82724 

182708 

49 
48 

13 

749987 

31  .0 
on  Q 

917462 

.0 

Uq 

832525 

45.3 

167475 

56232 

82692 

47 

14 

750172 

du.y 

917376 

.  O 
Uq 

832796 

XX  O 

167204 

56256 

82675 

46 

15 

750358 

30.9 

917290 

.  d 

833068 

4O  .  d 

166932 

56280 

82659 

45 

16 

750543 

30.9 

917204 

14.  3 

833339 

45.2 

166661 

56305 

82643 

44 

17 

750729 

30.9 

917118 

14.3 

833611 

45  .2 

166389 

56329 

82626 

43 

18 

750914 

30.9 

917032 

14.4 

833882 

45  .2 

166118 

56353 

82610 

42 

19 

751099 

30.8 

916946 

14.4 

834154 

45  .2 

165846 

56377 

82593 

41 

20 
21 

22 

751284 
9.751469 
751654 

30.8 
30.8 
30.8 

916859 
9.916773 
916687 

14.4 
14.4 
14.4 

834425 
9.834696 
834967 

45  .2 
45.2 
45.2 

165575 
10.165304 
165033 

56401 
56425 
56449 

82577 
82561 
82544 

40 
39 

38 

23 

751839 

30.8 

qA  Q 

916600 

14.4 
144 

835238 

45.2 

164762 

56473 

82528 

37 

24 
25 

752023 
752208 

ol>  .0 

30.7 

916514 
916427 

14.4 

14.4 

HA 

835509 
835780 

45.2 

164491 
164220 

56497 
56521 

82511 

82495 

36 
35 

26 

752392 

30  .  7 

916341 

,  4 

836051 

4o.l 

163949 

56545 

82478 

34 

27 

752576 

30.7 

916254 

14.4 

836322 

45.1 

A  K 

163678 

56569 

82462 

33 

28 

752760 

30.7 

916167 

14.4 

836593 

4o. 

163407 

56593 

82446 

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29 

752944 

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916081 

14.5 

836864 

45  . 

163136 

56617 

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753128 

30.6 

915994 

14.5 

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9.753312 

30.6 

9.915907 

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5(5065 

82396 

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753495 
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30.6 

915820 
915733 

14.5 
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837675 
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162325 
162054 

56689 
56713 

82380 
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753862  *"'* 

915646 

14.5 

1/1  K 

838216 

45.1 

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161784 

56736 

82347 

26 

35 

754046  IK'S 

915559 

14.  O 

He 

838487 

4o  .  1 

AK   f\ 

161513 

56760 

82330 

25 

36 

754229  JX  K 

915472 

.  o 

He 

838757 

40  .  0 

161243 

56784 

82314 

24 

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754412  30-5 

915385 

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839027 

45  .0 

160973 

56808 

82297 

23 

38 

754595  ^  £ 

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839297 

45  .  0 

160703 

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754778^'? 

915210 

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160432 

56856 

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754960  ,X'' 

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841187 

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914598 
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14.6 
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841457 
841726 

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44.9 

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158543 
158274 

57024 
57047 

82148 
82132 

14 
13 

48 

756418  fX  3 

914422 

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Uc 

841996 

44.  y 
44  9 

158004 

57071 

82115 

12 

49 

766600  IX  '1 

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157734 

57095 

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756782  iq"'  £ 

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A  A  Q 

10.157195 

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156926 

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4 

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768050  fX, 

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A  A  a 

155580 

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758230  |X'} 

913541 

14.7 

Hry 

844689 

44.  a 

44  fi 

155311 

57310 

81949 

2 

59 

758411  JX  t 

913453 

.  / 

844958 

44  ,  o 

A  A   Q 

155042 

57334 

81932 

1 

60 

758591  dU>1 

913365 

14.7 

845227 

44.  o 

154773 

57358 

81915 

0 

~ 

Cosine,  i 

S5neT~ 

Cotang. 

Tang. 

N.  cos. 

\  sine. 

55  Degrees. 

56          Log.  Sines  and  Tangents.  (35°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  1U' 

Cosine. 

D.  lu" 

Tang. 

D.  10" 

Cotang.  j  N.  sine. 

N.  cos. 

o 

9.758591 

9.913365 

.845227 

0.154773 

57358 

31915 

60 

1 

758772 

30.1 

913276 

[4.7 

845496 

44.8 

A  4   U 

154504 

57381 

31899 

59 

2 

758952 

30.0 

913187 

[4.7 

845764 

44.0 

A  A   G 

154236 

57405 

31882 

58 

3 

759132 

30.0 

913099 

14.8 

846033 

44.0 

A  A   Q 

153967 

57429 

31865 

57 

4 

759312 

30  0 

913010 

14.  8 

UQ 

846302 

44.0 

A  A   Q 

153698 

57453 

31848 

56 

5 

759492 

30.  v 

912922 

.0 

Uo 

846570 

44.0 

A  4   1 

153430 

57477 

31832 

55 

6 

759672 

30.0 

912833 

.0 
Uo 

846839 

44.  / 

153161 

57501 

31815 

54 

7 

759852 

29  .  9 

912744 

.0 

847107 

44.7 

152893 

57524 

31798 

53 

8 

760031 

29.9 

912655 

14.8 

-1  A   tt 

847376 

44.7 

152624 

57548 

81782 

52 

9 
10 

760211 
760390 

29.9 
29.9 

912566 
912477 

14.0 

14.8 

Uo 

847644 
847913 

44.7 
44.7 

Ail 

152356 
152087 

57572 
57596 

81765 

81748 

51 

50 

11 

9.760569 

29  .  9 

9.912388 

.0 
Uo 

9.848181 

41.  / 

10.151819 

57619 

81731 

49 

12 

760748 

29.  o 

912299 

.0 

848449 

44.7 

151551 

57643 

81714 

48 

13 

7(50927 

29.8 

orv  Q 

912210 

14.9 

848717 

44.7 

151283 

57667 

81698 

47 

14 

761106 

29.0 
c%n  Q 

912121 

14.9 

i  A  n 

848986 

44.7 

151014 

57691 

81681 

46 

15 

761285 

29.0 

912031 

14.9 
U(\ 

849254 

44.7 

150746 

57715 

81664 

45 

16 

761464 

29.8 

911942 

.y 

849522 

44.7 

150478 

57738 

81647 

44 

17 

761642 

29.8 

911853 

14.9 

849790 

44.7 

A  i  £1 

150210 

57762 

81631 

43 

18 
19 

761821 
761999 

29.7 
29.7 

911763 
9U674 

14.9 
14.9 

U(\ 

850058 
850325 

44.6 
44.6 

149942 
149675 

57786 
57810 

81614 
81597 

42 
41 

20 

762177 

29.7 

911584 

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850593 

44.6 

A  A   £Z 

149407 

57833 

81580 

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21 

9.762356 

29.7 

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14.9 
Ur\ 

9.850861 

44.D 

10.149139 

57857 

81563 

39 

22 
23 
24 

762534 
762712 
762889 

29.7 
29.6 
29.6 

911405 
911315 
911226 

,9 
14.9 
15.0 
1  PC  A 

851129 
851396 
851664 

44.6 
44.6 
44.6 

A  A   (Z 

148871 
148604 
148336 

57881  81546 
6790481530 
5792881513 

38 
37 
36 

25 

763067 

ay.  6 

911136 

lo.  u 

1  K  A 

851931 

44.  0 

148069 

57952 

81496 

35 

26 

763245 

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911046 

lo.  u 

1  K  A 

852199 

44.6 

A  A  (^ 

147801 

57976 

81479 

34 

27 

763422 

29.6 

910956 

lo.  U 
1  K  A 

852466 

44.  D 

A  A  fi 

147534 

57999  81462 

33 

28 

763600 

29.6 

910866 

lo  .  0 

852733 

44.  o 

147267 

58023 

81445 

32 

29 

763777 

29.5 

910776 

15.0 

i  K  A 

853001 

44.5 

A  A   K 

146999 

58047 

81428 

31 

30 
31 

763954 
9.764131 

29^5 

910686 
9.910596 

10  .  U 

15.0 

853268 
9.853535 

44  .  0 

44.5 

146732 
10-146465 

5807081412 
5809481395 

30 

29 

32 

764308 

29.5 

910506 

15.0 

853802 

44.5 

146198 

5811881378 

28 

33 

764485 

29.5 

910415 

15.0 

854069 

44.5 

A  A   K 

145931 

58141 

81361 

27 

1  34 

764662 

29.4 

910325 

15.0 

854336 

44.  0 

145664 

58165 

81344 

26 

35 

764838 

29.4 

910235 

15.1 

854603 

44.5 

145397 

58189 

81327 

25 

36 

765015 

29.4 

910144 

15.1 

854870 

44.  5 

145130 

58212 

81310 

24 

37 

765191 

29.4 

910054 

15.1 

855137 

44.5 

144863 

58236 

81293 

23 

38 

765367 

29.4 

909963 

15.1 

855404 

44.  5 

144596 

58260 

81276 

22 

39 

765544 

29.4 

909873 

16.1 

855671 

44.5 

144329 

58283 

81259 

21 

40 

765720 

29.3 

909782 

15.1 

855938 

44.4 

144062 

58307 

81242 

20 

41 

9.765896 

29.3 

9.909691 

15.1 

9.856204 

44.4 

10-143796 

58330 

81225 

19 

42 

766072 

29.3 

909601 

15.1 

856471 

44  .4 

143529 

58354 

81208 

18 

43 

766247 

29.3 

909510 

15.  1 

856737 

44.4 

143263 

58378 

81191 

17 

44 

766423 

29.3 

909419 

15.1 

857004 

44.4 

142996 

58401 

81174 

16 

45 

766598 

29.3 

909328 

15.1 

857270 

44.4 

142730 

58425 

81157 

15 

46 

766774 

29.2 

909237 

15.2 

857537 

44.4 

142463 

58449 

81140 

14 

47 

766949 

29.2 

909146 

15.2 

857803 

44.4 

142197 

58472 

81123 

13 

48 

767124 

29.2 

909055 

15.2 

858069 

44.4 

141931 

58496 

81106 

12 

49 

767300 

29.2 

908964 

15.2 

858336 

44.4 

141664 

58519 

81089 

11 

50 

767475 

29.2 

908873 

15. 

858602 

44.4 

141398 

58543 

81072 

10 

61 

9.767649 

29.1 

9.908781 

15. 

9.858868 

44.; 

10-141132 

58567 

81055 

9 

52 

767824 

29.1 

908690 

15.' 

859134 

44,; 

140866 

58590 

81038 

8 

53 

767999 

29.  1 

908599 

15.- 

859400 

44J 

140600 

58614 

81021 

7 

54 

768173 

29.  1 

908507 

15.' 

859666 

44.J 

140334 

58637 

81004 

6 

55 

768348 

29.1 

908416 

15. 

859932 

44.; 

140068 

58681 

80987 

5 

56 

768522 

29.  C 

908324 

15.; 

860198 

44.; 

139802 

58684 

80970 

4 

67 

768697 

29.  C 

908233 

15.  c 

860464 

44.; 

139536 

58708 

80953 

3 

58 

768871 

29.0 

908141 

15.^ 

861)730 

44.; 

139270 

58731 

80936 

2 

59 

769045 

29.0 

908049 

15.  J 

860995 

44.; 

139005 

58755 

80919 

1 

60 

769219 

29.0 

907958 

16. 

861261 

44.; 

138739 

58779 

80902 

0 

Cosine. 

Sine. 

Cotang. 

Tang. 

N.  cos. 

N.sine 

' 

54  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (36°)  Natural  Sines. 

57 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.   N.  sint;. 

N.  cos. 

0 
1 

o 

9.769219 
769393 
769566 

29.0 

28.9 

9.907958 
907866 
907774 

15.3 
15.3 

9.861261 
861527 
861792 

44.3 
44.3 

10.138739J  58779 
138473  J58802 
138208  58826 

80902 
80885 
80867 

60 
59 

58 

3 
4 

769740 
769913 

28.9 
28.9 

907682 
907590 

15.3 
15.3 

86203S 
862321. 

44  2 
4-i.  2 

1  37942  i!  58849:80850 
137677  5887380833 

57 
56  j 

5 

770087 

28.9 
98  Q 

907498 

15.3 

1  e  q 

862589 

44.2 

Ail  O 

137411 

58896  80816 

55 

6 

770260 

^o  .  y 

90  Q 

907406 

10  .  o 
1  e.  q 

862854 

i±'±  .  Z 
A  A  O 

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58920  80799 

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10.  4 

i  p-  A 

864445 

44  .  Z 
A  A  O 

135555 

59061 

80696 

48 

13 
14 

771470 
771643 

-*-o  .  / 

28.7 
OQ  n 

906760 
906667 

1O.4 

15.4 

864710 
864975 

44  .  Z 

44.2 

135290 
135025 

59084  80679 
59108180662 

47 
46 

15 

771815 

~!o.  7 
OQ  n 

906575 

15.4 

865240 

44. 

134760 

59131J80644 

45 

16 

771987 

iio.7 

OQ  o 

906482 

15.4 

1C  A 

865505 

44. 

134495 

6915480627 

44 

17 

772159 

^o.7 

OU  rf 

906389 

15.4 

IK   K. 

865770 

44. 

134230 

5917880610 

43 

18 

772331 

2o.7 

906296 

15.  o 

1  K   K 

866035 

44. 

133965 

59201 

80593 

42 

19 

772503 

28.  6 
OQ  r* 

906204 

15.  o 

1  p-  e 

866300 

44. 

A  A 

133700 

59225 

80576 

41 

20 

772675 

Zfj  .  (j 

oQ  r* 

906111 

10  .  o 

1  {-   K 

866564 

44. 

133436 

59248 

80558 

40 

21 

9.772847 

ZV>  .  O 
OQ  f> 

9.906018 

15.  o 

IK   Z. 

9.866829 

44  . 

A  A 

10.133171 

59272  80541 

39 

22 

773018 

Zo  .  D 

905925 

IO»" 

867094 

44  . 

132906  59295 

80524 

38 

23 

24 

773190 
773361 

28.6 
28.6 

98  Pi 

905832 
905739 

15.5 
15.5 
1^5 

867358 
867623 

44. 
44. 

132642 
132377 

5931880507 
5934280489 

37 
36 

25 

773533 

•*o  .  o 

OQ  •* 

905645 

10  .  o 

IK   K. 

867887 

44. 

132113 

59365 

80472 

35 

26 

773704 

•*o  .  O 
OQ  e 

905552 

lo  .  o 

1  •*  p^ 

868152 

44.1 

131848 

59389 

80455 

34 

27 
28 
29 
30 

773875 
774046 
774217 

774388 

ijo.5 

28.5 
28.5 
28.5 

905459 
905366 
905272 
905179 

lo.o 
15.5 
15.6 
15.6 

868416 
868680 
868945 
869209 

44.0 
44.0 
44.0 
44.0 

131584 
131320 
131055 
130791 

5941280438 
59436:80422 
5945980403 

59482180386 

33 
32 
31 
30 

31 

9.774558 

28.4 

OQ  A 

9.905085 

15.6 
i  -  fi 

9.869473 

44.0 
A  \  n 

10.130527  5950680368 

29 

32 

774729 

•^O  .  4 
OQ  A 

904992 

10  .  u 
1  K.  R 

869737 

44  .  U 

130263 

59529|80351 

28 

33 

774899 

£o  ,  4 

OQ  A 

904898 

lo.o 

1  K  f\ 

870001 

44.0 

129999 

5955280334 

27 

j  34 

775070 

Zo  .4 

OQ  A 

904804 

15.  D 
1  K  K 

870265 

44.0 

129735 

5957680316 

26 

35 

775240 

Ho  .4 
98  A 

904711 

15  .  o 
•IK  f; 

870529 

44.0 

129471 

5959980299 

25 

36 

775410 

Zo  .  4 

OQ  O 

904617 

iO  .  O 
1  -  c 

870793 

44  .  0 

A  A  A 

129207 

5962280282 

24 

37 

775580 

^o  ,  o 

OQ  O 

904523 

IO  .  D 

If  P 

871057 

44.  U 

128943 

59646 

80264 

23 

38 

775750 

2o  .  d 

904429 

o.o 

1^  *y 

871321 

44.0 

128679 

59669 

80247 

22 

39 

775920 

28.3 

OQ  Q 

904335 

0.  / 
1  -  o 

871585 

44.0 

128415 

59693 

80230 

21 

40 

776090 

~o.  o 

OQ  0 

904241 

lo.  / 

1  n  7 

871849 

44.0 

A<1   Q 

128151 

59716 

80212 

20 

41 

9.776259 

Zo  .  o 

OQ  q 

9.904147 

J  O  .  < 

1  n  7 

9.872112 

*o  .y 

/!*}  Q 

10.127888 

59739 

80195 

19 

42 

776429 

Zo  .  O 

9,8  9 

904053 

IO  .  I 

I  ^  7 

872376 

4o.y 

127624 

69763 

80178 

18 

43 
44 

776598 
776768 

-^o  .  Z 
28.2 

OQ  O 

903959 
903864 

10.1 

15.7 

1  PI  7 

8  i  2640 
872903 

43  .9 
43.9 

,1Q  O 

127360 
127097 

5978680160 
6980980143 

17 
16 

45 

776937 

Zo  .  Z 

903770 

10.4 

1r  ^ 

873167 

4o  .y 

126833 

59832 

80125 

15 

46 

777106 

28.2 
28  9, 

903676 

0.  / 

1^7 

873430 

43.9 

4q  q 

126570 

59856 

80108 

14 

47 

777275 

z&  .  z 

903581 

IO  ,  I 

873694 

4o  .  y 

126306 

59879 

80091 

13 

48 
49 

777444 
777613 

28.  1 
28.1 
98  1 

903487 
903392 

15.7 
15.7 

1  K   C 

873957 
874220 

43.9 
43.9 

A  0   Q 

126043 
125780 

!  59902180073 
!  59926  80056 

12 
11 

50 

777781 

^o  .  1 

28  1 

903298 

1O.O 

15  8 

874484 

4<_>  .  y 

40  Q 

125516 

59949180038 

10 

51 

9.777950 

9.903202 

1"  Q 

9.874747 

4o  .  J 

10.125253 

69972 

80021 

9 

52 
53 
54 
55 
56 
57 

778119 
778287 
778455 
778624 
778792 
778960 

28.  1 
28.1 
28.0 
28.0 
28.0 
28.0 

OQ  A 

903108 
903014 
902919 
902824 
902729 
902634 

O.o 

15.8 
15.8 
15.8 
15.8 
15.8 

1  ^  Q 

875010 
875273 
875536 
875800 
876063 
876326 

43.9 
43.9 

43.8 
43.8 
43.8 
43.8 

A  O   Q 

124990 
124727 
124464 
124200 
123937 
123674 

59995 
|  60019 
j  60042 
!  60065 
!  60089 
160112 

80003 
79986 
79968 
79951 
79934 
79916 

8 
7 
6 
5 
4 
3 

58 

779128 

Zo.  0 

902539 

lo  .  o 

876589 

4d.O 

123411  60135 

79899 

2 

59 

779295 

28  .  0 

902444 

15  .9 

876851 

43.8 

123149 

60158 

79881 

1 

60 

779463 

27  .  9 

902349 

15.9 

877114 

43.8 

122886  60182 

79864 

0 

Cosine. 

S7ne7~ 

Co  tang. 

Tang.    '  N.  cos 

N.sine. 

' 

53  Degrees. 

58          Log.  Sines  and  Tangents.  (37°)  Natural  Sines.     TABLE  II. 

i 

Sine. 

D.  10 

Cosine.  |D.  10 

Tang. 

D.  10' 

'  Cotang.  ||N  .sine 

N.  cos 

0 

9.779463 

27  ^ 

9.902349 

i  K  n 

9.877114 

10.122886 

60182 

79864 

60 

1 

779631 

902253 

15.  £ 

IK  n 

877377 

43.8 

122623 

60205 

79846 

59 

2 

779798 

27  .  £ 
97  Q 

902158 

15.  £ 
1  K   Q 

877640 

43.8 

122360 

60228 

79829 

58 

3 

779966 

•61  ,y 

902063 

lo.y 

1  K  n 

877903 

4d.c 

122097 

60251 

79811 

57 

4 

780133 

97  Q 

901967 

15.9 
i  K  Q 

878165 

43.8 

121835 

60274 

79793 

56 

5 

780300 

M  /  .  y 

97  8 

901872 

10.  y 

1  K  C 

878428 

,40  Q 

121572 

60298 

79776 

55 

6 

780467 

•*  /  .  o 
97  S 

901776 

10.  y 

-i  K  r 

878691 

4d  .  O 

121309 

60321 

79758 

54 

7 

780634 

£  1  ,  O 

OT  O 

901681 

lo.b 

878953 

43.8 

121047 

60344 

79741 

53 

8 

780801 

21  .  o 
97  R 

901585 

15.9 

1  ^  Q 

879216 

43.7 

120784 

160367 

79723 

52 

9 

780968 

41  .0 

901490 

lo.  y 

IK  Q 

879478 

A3  ' 

120522|i60390 

79706 

51 

10 

781134 

c*n  *  o 

901394 

lo  .  a 

879741 

4d. 

120259 

60414179688 

50 

11 

9.781301 

27.o 

9.901298 

16.  C 

9.880003 

43.' 

10.119997 

60437 

79671 

49 

12 

781468 

27.7 

901202 

16.  C 

880265 

43.' 

119735 

60460 

79658 

48 

13 

781634 

27.7 

901106 

16.0 

880528 

43.' 

119472 

60483 

79635 

47 

14 

781800 

27.7 

901010 

16.0 

880790 

43. 

119210 

60506 

79618 

46 

15 

781966 

27.7 

900914 

16.0 

881052 

43. 

118948 

60529 

79600 

45 

16 

782132 

27.7 

900818 

16.0 
1  n  n 

881314 

43.' 

118686 

60553 

79583 

44 

17 

782298 

27.7 

900722 

lo.  U 

f  r»  *  f\ 

881576 

43.  ' 

118424 

60576 

79565 

43 

18 

782464 

27.6 

r\r*  r; 

90&626 

lo.  U 

1  r*  '  f\ 

881839 

43  .' 

118161 

60599 

79547 

42 

19 

782630 

27.  O 

900529 

lo.  U 
i  r*  "  A 

882101 

43  .  ' 

117899 

60622 

79530 

41 

20 

782796 

27.6 

900433 

lo.  U 

882363 

43.  ' 

117637 

60645 

79512 

40 

21 

9.782961 

27.6 

9.900337 

16.1 

9.882625 

43.6 

10.117375 

60668 

79494 

39 

22 

783127 

27.6 

900242 

16.  1 

882887 

43.6 

117113 

60691 

79477 

38 

23 

783292 

27.6 

900144 

16.1 

883148 

43.6 

116852 

60714 

79459 

37 

24 

783458 

27.5 

900047 

16.  1 

883410 

43.6 

116590 

60738 

79441 

36 

25 

783623 

27.5 

899951 

16.1 

883672 

43.6 

1  16328 

60761 

79424 

35 

26 

783788 

27.5 

899854 

16.1 

883934 

43.6 

116066 

60784 

79406 

34 

27 

783953 

27.5 

899757 

16.1 

884196 

43.6 

115804 

6080? 

79388 

33 

28 

784118 

27.5 

899660 

16.1 

884457 

43.6 

115543 

60830 

79371 

32 

20 

784282 

27.5 

899564 

16.  1 

884719 

43.6 

115281 

60853 

79353 

31 

39 

784447 

27.4 

899467 

16.1 

884980 

43.6 

115020 

60876 

79835 

30 

31 

9.784612 

27.4 

9.899370 

16.2 

9.885242 

43.6 

10.H4758 

60899 

79318 

29 

32 

784776 

27.4 

899273 

16.2 

885503 

43.6 

114497 

60922 

79300 

28 

33 

784941 

27.4 

899176 

16.2 

885765 

43.6 

114235 

60945 

79282 

27 

34 

785105 

27.4 

899078 

16.2 

886026 

43.6 

113974 

60968 

79264 

26 

35 

785269 

27.4 

898981 

16.2 

886288 

43.6 

113712 

60991 

79247 

25 

36 

785433 

27.3 

898884 

16.2 

886549 

43.6 

113451 

61015 

79229 

24 

37 

785597 

27.3 

898787 

16.2 

886810 

43.5 

113190 

61038 

79211 

23 

38 

785761 

27.3 

898689 

16.2 

887072 

43.  5 

112928 

61061 

79193 

22 

39 

785925 

27.3 

898592 

16.2 

887333 

43.5 

112667 

61084 

79176 

21 

40 
41 

42 

786089 
.  786252 
786416 

27.3 
27.3 

27.2 

898494 
9.898397 
898299 

16.2 
16.3 
16.3 

887594 
9.887855 
888116 

43.5 
43.5 
43.5 

A  O  C 

112406 
10.112145 
111884 

61107  79158 
6113079140 
6115379122 

20 
19 
18 

43 

786579 

27.2 

898202 

16.3 

888377 

4o  .  5 

111623 

61176 

79105 

17 

44 

786742 

27.2 

898104 

16.3 

888639 

43  .  5 

111361 

61199 

79087 

16 

45 

786906 

27.2 

O'Y  O 

898006 

16.3 

1  d  O 

888900 

43  .5 

A  Q  K 

iinoo 

61222 

79069 

15 

46 

787069 

21  ,2 

897908 

Lo  .  d 

889160 

4d.O 
A  Q  K 

11  0840  i 

61245 

79051 

14 

47 

787232 

27.2 

897810 

L6.  3 

889421 

4o  .  O 

110579' 

61268 

79033 

13 

48 

787395 

27.  1 

897712 

[6.  3 

889682 

43  .  5 

110318 

61291 

79016 

12 

49 

787557 

27.1 

897614 

[6.3 

889943 

43.5 

110057 

61314 

78998 

11 

50 

787720 

27.  1 

897516 

I6.3 

890204 

To 

109796  ' 

6133? 

78980 

10 

51 

.787883 

27.  1 

3.897418 

L6.3 

.890465 

4d.  4 

0.109535 

61360 

78962 

9 

52 
53 

788045 

788208 

27.1 
27.1 

897320 
897222 

16.4 

L6.4 

890725 
890986 

43.4 
43.4 

JO   A 

1U9275  61383 
109014:  61406| 

78944 
78926 

8 

7 

54 
55 

788370 
788532 

27.  1 
27.0 

897123 
897025 

^6.4 
16.4 

891247 
891507 

T:<->  .  4 

43.4 

108753  !  61429 
108493  161451 

78908 
78891 

6 
5 

66 

788694 

27.0 

896926 

16.4 

6      A 

891768 

43.4 

108232  j 

61474' 

78873 

4 

57 

788856 

27.0 

896828 

.  4 

892028 

43.4 

107972  i 

61497  ' 

"8855 

3 

68 

789018 

27.0 

896729 

6.4 

892289 

4d.4 

107711 

61520 

f8837 

2 

59 

789180 

27.0 

896631 

6.4 

892549 

43.4 

107451 

61543 

C8819 

1 

60 

789342 

27.0 

896532 

6.4 

892810 

43  .  4 

107190 

61566  ' 

"8801 

0 

Cosine. 

Sine. 

Cotang. 

Tang.    IX  cos. 

M.sine. 

' 

52  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (38°)  Natural  Sines.          59. 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.  10" 

Cotang.  j  N.  sine.jN.  cos. 

0 

9.789342 

9.896532 

9.892810 

10.107190 

61566 

78801 

60 

1 

789504 

26.9 

896433 

16.4 

893070 

43.4 

106930 

61589 

78783 

59 

2 

789665 

26.9 

896335 

16.5 

893331 

43.4 

106669 

61612 

78765 

58 

3 

789827 

26.9 

896236 

16.5 

893591 

43.4 

10b409 

61635 

78747 

57 

4 

789988 

26.9 

896137 

16.5 

893851 

4'  :  .  4 

106149 

61658 

78729 

56 

6 

790149 

26.9 

896038 

16.5 

894111 

43.4 

105889 

61681 

78711 

55 

6 

790310 

26.9 

895939 

16.  5 

894371 

43.4 

105629 

61704 

78694 

7 

790471 

26.8 

895840 

16.6 

1  ft  K 

894632 

43.4 

105368 

61726 

78676 

53 

8 

790632 

26.8 

895741 

Ib.o 

1  ft  K 

894892 

43.3 

105108 

61749 

78668 

52 

9 

790793 

26.8 

895641 

Ib.  o 

895152 

43.3 

104848 

61772 

78640 

51 

10 

790954 

26.8 

895542 

16.  5 

895412 

43.3 

104588 

61795 

78622 

50 

11 

9.791115 

Oft  Q 

9.895443 

16.5 

9.895672 

43.3 

10.104328 

61818 

78604 

49 

12 

791275 

2b  .0 
oft  ry 

895343 

16.6 

1  ft  ft 

895932 

43  .3 

104068 

61841 

78586 

48 

13 

791436 

2b  .7 

895244 

Ib.b 
1  ft  ft 

896192 

43.3 

103808 

61864 

78568 

47 

14 

791596 

26  .  7 

895145 

Ib.b 
1  ft  ft 

896452 

43.3 

103548 

61887 

78550 

46 

15 

791757 

26  .7 

895045 

Ib.b 

896712 

43  .3 

103288 

61909 

78532 

45 

16 

791917 

*?" 

894945 

16.6 

896971 

43.3 

103029 

61932 

78514 

44 

17 

792077 

26  .  7 

894846 

16.6 

897231 

43  .3 

102769 

61955 

78496 

43 

18 

792237 

26  .7 

Oft  ft 

894746 

16.6 

1  £?   ft 

897491 

43  .3 

102509 

61978 

78478 

42 

19 

792397 

26.  b 

894646 

16.  b 

897751 

43-3 

102249 

62001 

78460 

41 

20 

792557 

26.6 

894546 

16.6 

898010 

43.3 

101990 

62024 

78442 

40 

21 

9.792716 

26.6 

Oft  £? 

9.894446 

16.6 

9.898270 

43-3 

10.101730 

62046 

78424 

39 

22 

792876 

2b  .b 

894346 

16.7 

898530 

43.3 

101470 

62069 

78405 

38 

23 

793035  |™'2 

894246 

16.7 

Ifi  7 

898789 

43.3 

40  o 

101211 

62092 

78387 

37 

24 

793195 

*  •" 

894146 

ID  .  ' 

899049 

100951 

62115 

78369 

36 

25 

793354 

26.5 
26  5 

894046 

16.7 
16  7 

899308 

43.2 

100692 

62138 

78351 

36 

26 

793514 

893946 

1  ft  i 

899568 

xo 

100432 

62160 

78333 

34 

27 

793673 

26  .  6 

Oft  K. 

893846 

16.  / 

1  ft  *7 

899827 

43  .2 

100173 

62183 

78315 

33 

28 

793832 

*b  .0 

893745 

16.  i 

1  ft  "7 

900086 

43  .2 

099914 

62206 

78297 

32 

29 

793991 

26.5 

Oft   E 

893645 

16.  t 

1ft  7 

900346 

40  9 

099654 

62229 

78279 

31 

30 

794150 

-*O  .  O 
Oft  /I 

893544 

ID  .  / 
1  ft  1 

900605 

Ho  .  Z 

099395 

62251 

78261 

30 

31  |9.  794308 

xib  .4 

Oft  A 

9.893444 

Ib.  t 

1ft  Q 

9.900864 

43  .2 
43  ^ 

10.099136 

62274 

78243 

29 

32 

794467 

oft  4 

893343 

ID  .  o 

1  ft  ft 

901124 

098876 

62297 

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893142 

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901642 

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098358 

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794942  I*"" 

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098099 

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892940 

ID  .  o 
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902160 

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40  9 

097840 

62388 

78152 

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796359  ~rj 

892839 

ID  .  o 
1  (  \  H 

902419 

Ho  .  Z 
40  9 

097581 

62411 

78134 

23 

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796417  IS  a 

892739 

ID  .  o 

1ft  C 

902679 

Ho  .  Z 
40  9 

097321 

62433 

78116 

22 

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795575  £J'a 

892638 

ID  .  o 
1  ft  ft 

902938 

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40  9 

097062 

62456 

78098 

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892536 

ID  .  o 

1  ft  Q 

903197 

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096803 

62479 

78079 

20 

41 

9.  795891  if?^ 

9.892435 

ID  .  o 
1  ft  Q 

9.903455 

40  i 

10.096545 

62502 

78061 

19 

42 

796049  f?'f 

892334 

ID  .  y 

903714 

HO  .  1 
40  -i 

096286 

62524 

78043 

18 

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796206  ,£'* 

892233 

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1  ft  Q 

903973 

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40  i 

096027 

62547 

78025 

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796864  S'! 

892132 

ID  .  y 

1  ft  O 

904232 

HO  .  1 
40  i 

095768 

62570 

78007 

16 

45 

796521  IT* 

892030 

ID  .  y 

904491 

HO  .  1 

A  Q   . 

095509 

62592 

77988 

15 

46 

47 
48 

796679  g'J 
796836  2'J 
796993  fj  o 

891929 
891827 
891726 

16.9 
16.9 
16.9 

904750 
905008 
905267 

Hd.l 

43.1 
43.1 

095250 
094992 
094733 

62615 
62638 
62660 

77970 
77952 
77934 

14 
13 
12 

49 

797150  £** 

891624 

16.9 

905526 

43.1 

094474 

62683 

77916 

11 

50 

797307  £*•* 

891523 

16.9 

905784 

43.  1 

094216 

62706 

77897 

10 

51 

9.797464  *': 

9.891421 

17.0 

9.906043 

43.1 

10.093957 

62728 

77879 

9 

52 

797621  *°': 

891319 

17.0 

906302 

43.1 

093698 

62751 

77861 

8 

53 

797777  9^ 

891217 

17.0 
nfi 

906560 

43.1 

A  Q 

093440 

62774 

77843 

7 

54 

797934  oft' 

891115 

•  U 

906819 

HO  . 

093181 

62796 

77824 

6 

;  55 

798091  *': 

891013 

17.0 

907077 

43  . 

092923 

62819 

77806 

G 

56 

798247  *': 

890911 

17.0 

907336 

43  . 

092664 

62842 

77788 

4 

57 

798403  ~'* 

890809 

17.0 

907594 

43  . 

092406 

62864 

77769 

3 

58 

798560  fj!*U 

890707 

17.0 

nn 

907852 

43  . 
40 

092148 

62S87 

77751 

2 

59 

798716  JJ-J 

890605 

.  U 

908111 

HO  . 

091889 

62909 

77733 

1 

60 

798872  26'° 

890.303 

17.0 

908369 

43.0 

091631 

62932 

77715 

0 

Cosine. 

SiuR. 

Cotang. 

Tang. 

:M  .  cos". 

N.sine- 

r 

51  Degrees. 

60          Log.  Sines  and  Tangents.  (39°)  Natural  Sines.     TABLE  II. 
_ 

Sine. 

D.  10' 

Cosine. 

D.  10 

Tang. 

L>.  10 

Cotang. 

N.  sine 

N.  cos 

o 

9.798772 

9.890503 

9.903369 

10.091631 

62932 

77715 

60 

799028 

26.  C 

890400 

17.  t 
1  7  1 

908628 

43.  ( 

091372 

62955 

77696 

59 

<• 

799184 

9R  f 

890298 

1  /  .  1 
ni 

908886 

40  f 

091114 

62977 

77678 

58 

i 

799339 

~?  g 

890195 

.  1 
ni 

909144 

4t3  .  1. 
40  ( 

090856 

63000 

77660 

57 

i 

799495 

ORfl 

890093 

.  L 
ni 

909402 

4o  .  I. 
40  f 

090598 

63022 

77641 

56 

6 

799651 

o-  q 

889990 

.  1 
ni 

909360 

4o  .  (_ 
40  i 

090340 

63045 

77623 

55 

6 

799806 

Ofi'fl 

889888 

.  1 

1  7  1 

909918 

4o  .  L 

090082 

63068 

77605 

54 

799962 

2X  Q 

889785 

1  /  .  1 
17  1 

910177 

40  / 

089823 

63090 

77586 

53 

8 

800117 

o.y 

889682 

1  '  .  A 

910435 

4o  .  I. 

039565 

63113 

77568 

52 

9 

800272 

25.9 

OK  Q 

889579 

ni 

910693 

40  / 

089307 

63135 

77550 

51 

10 

800427 

ZO  .  O 
c*~  O 

889477 

.  J 

n-t 

910951 

4o  .  I. 

089049 

63158 

77531 

50 

11 

9.800582 

2o.o 

9.889374 

.  I 

9.911209 

43  .  C 

A  O  A 

10.088791 

93180 

77513 

49 

800737 

25.8 

OK  O 

889271 

17.' 

911467 

4o  .  I 

40   A 

038533 

63203 

77494 

48 

13 

800892 

zo.o 

889168 

, 

911724 

4o  .  I 

088276 

63225 

77476 

47 

14 

801047 

25.8 

889064 

17.x 

911982 

43.  C 

038018 

63248 

77458 

46 

15 

801201 

25.8 

888961 

17. 

912240 

43.  C 

/iO  A 

037760 

63271 

77439 

45 

16 

801356 

25.8 

888858 

. 

912498 

4d.O 

A  O   A 

087502 

63293 

77421 

44 

17 

801511 

25.7 

888755 

17. 

912756 

43.1 

087244 

63316 

77402 

43 

18 

801665 

25.7 

888651 

17. 

913014 

43.  C 

*O  A 

036986 

63338 

77384 

42 

19 

801819 

25.7 

888548 

17. 

913271 

42.9 

086729 

63361 

77366 

41 

20 

801973 

25.7 

888444 

17. 

913529 

42.9 

086471 

63383 

77347 

40 

21 

9.802128 

25.7 

9.888341 

17.3 

no 

9.913787 

42  .  <j 
42  9 

10-086213 

63406 

77329 

39 

22 

802282 

2o  •  7 

838237 

.  o 

914044 

035956 

63428 

77310 

38 

23 

802436 

25.6 

888134 

17.3 
17  3 

914302 

42.9 
42  ^) 

085698 

63451 

77292 

37 

24 

802589 

2o.b 

888030 

nr> 

914560 

085440 

63473 

77273 

36 

25 

802743 

25.6 

887926 

.O 

no. 

914817 

42  .  9 

42  9 

035183 

63496 

77255 

35 

26 

802897 

O"  ft 

887822 

.  O 

no 

915075 

42  9 

034925 

63518 

77236 

34 

27 

803050 

ZO-  D 

887718 

.0 

1  7   q 

915332 

4Q  A 

084668 

63540 

77218 

33 

28 

803204 

25.6 

887614 

J.  '  .  o 

n0 

915590 

4z  .  y 

084410 

63563 

77199 

32 

29 
30 

803357 
803511 

25.6 
25.5 

887510 
887406 

.3 

17.3 

HA 

915847 
916104 

42.9 
42.9 

084153 

083896 

63585 

63603 

77181 
77162 

31 
30 

31 

9.803664 

25.5 

9.887302 

.4 

9.916362 

42.  9 

10-083638 

63630 

77144 

29 

32 

803817 

25.5 

887198 

17.4 

nA 

916619 

42.9 

083381 

63653 

77125 

28 

33 

803970 

25.5 

887093 

.4 

nA 

916877 

42.9 

083123 

63675 

77107 

27 

34 

804123 

25-5 

886989 

.4 

917134 

42.9 

082866 

63698 

77088 

26 

35 

804276 

25-5 

886885 

17.4 

917391 

42.  9 

082609 

63720 

77070 

25 

36 

804428 

25-4 

886780 

17.4 

n4 

917648 

42.9 

/to  n 

082352 

63742 

77051 

24 

37 

804581 

25.4 

886676 

.  4 

nA 

917905 

4^.y 

082095 

63765 

77033 

23 

38 

804734 

25-4 

886571 

.  4 

918163 

42.9 

081837 

63787 

77014 

22 

39 

804886 

25.4 

886466 

17.4 

ni 

918420 

42.8 

40  c 

081580 

63810 

76996 

21 

40 

805039 

25-4 

OK  A 

886362 

.  4 
nt 

918677 

z  .  o 

081323 

63832 

76977 

20 

41 
42 

9.805191 
805343 

ZO  -4 

25.4 

9-886257 
886152 

.  o 
17.5 

9.918934 
919191 

42.'  8 

10-081066  i  63854 
0808091163877 

76959 
76940 

19 

18 

43 

805495 

25-3 

886047 

17.5 

919448 

•  o 

080552 

63899 

76921 

17 

44 

805647 

25.3 

885942 

17  .  5 

919705 

42.8 

080295 

63922 

76903 

16 

45 

805799 

25.3 

885837 

17  .  5 

919962 

42.8 

080038 

63944 

76884 

15 

46 
47 

48 

805951 
806103 
806254 

25.3 
25.3 
25.3 

885732 
885627 
885522 

17.5 
17.5 
17.5 

920219 
920476 
920733 

42.8 
42.8 
42.8 

079781 
079524 
079267 

63966  76866 
63989(76847 
6401176828 

14 
13 
12 

49 

806406 

25.3 

885416 

17.5 

nK 

920990 

42.8 

079010  |  64033 

76810 

11 

50 

806557 

zo  .  z 

885311 

.  o 

921247 

4z.  o 

078753  64056 

76791 

10 

51 

.806709 

25.2 

9.835205 

17.6 

.921503 

42.8 

0-078497 

64078 

76772 

9 

52 
53 

806860 
807011 

25.2 

25.2 

885100 
884994 

17.  6 
17.6 

921760 
922017 

42.'  8 

078240 
077983  j 

64100 
64123 

76754 
76735 

8 

7 

54 

807163 

25.2 

884889 

17.6 

9222/4 

42.8 

077726  64145 

76717 

6 

55 

807314 

25.2 

884783 

17.6 

922530 

42.8 

077470  64167 

76698 

5 

56 
57 

58 

807465 
807615 
807766 

25.2 
25.1 
25.1 

884677 
884572 
884466 

17.6 
17.6 
17.6 

92278  7 
923044 
92330J 

42  »8 
42.8 
42.8 

077213l!64190 
076956  64212 
076700'  64234 

76679 
76661 

76642 

4 
3 

2 

59 

807917 

25.  1 

17.6 

923557 

42.8 

076443  ;  64256 

76623 

1 

60 

808067 

25.  1 

884254 

L7.6 

923813 

42.7 

076187  jj  64279 

"6604 

0 

Cosine. 

Sine. 

Cotang. 

Tang.    N.  cos. 

S.sinc. 

50  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (40°)  Natural  Sines.          61 

"7 

Sine. 

D.  10" 

Cosine.  |D.  10" 

Tang. 

D.  10" 

Cotang. 

N  .sine. 

N.  cos. 

0 

9.808067 

9.884254 

177 

9.923813 

10.076187 

64279 

76604 

60 

1 

808218 

of  i 

884148 

1  /  .  / 

924070 

42.7 

075930 

64301 

76586 

59 

2 

808368 

S)K  1 

884042 

17.7 

177 

924327 

42.  7 

AO  7 

075673 

64323 

76567 

58 

3 

808519 

>-'>  ,  1 

883936 

1  '  .  ' 

924583 

4^5  .  / 

075417 

64346 

76548 

57 

4 

808669 

25  ,  0 

883829 

17.7 

924840 

42.  7 

075160 

64368 

76530 

56 

5 

808819 

o-  n 

883723 

17.7 

925098 

42.7 

074904 

64390 

76511 

55 

6 

7 

808969 
809119 

25^0 

O"'   A 

883617 
883510 

17.7 
17.7 

925352 
925609 

42.  7 
42.7 

074648 
074391 

64412 
64435 

76492 
76473 

54 
53 

1  8 

809269 

zo  ,  0 

883404 

17.7 

nm 

925865 

42.7 

074135 

64457 

76455 

52 

9 

809419 

O   Q 

883297 

.  1 
no 

926122 

42.  / 

073878 

64479 

76436 

51 

10 

809569 

24.9 

883191 

.8 

926378 

42.7 

073622 

64501 

76417 

50 

11 

9.809718 

24.  9 

OA  Q 

9.883084 

17.8 

no 

9.926634 

42.7 

AO  fjf 

10.073366 

64524 

76398 

49 

12 

809868 

^4,  y 

882977 

.0 

926890 

4^  ,  I 

073110 

64546 

76380 

48 

13 

810017 

24,9 
24  9 

882871 

17.8 

no 

927147 

42.7 

072853 

64568" 

76361 

47 

14 

810167 

O  A  O 

882764 

.0 

927403 

42.7 

072597 

64590 

76342 

46 

15 

810316 

24,  y 

018 

882657 

17.8 

927659 

42.  7 

072341 

64612 

76323 

45 

16 

810465 

24.O 
OA  8 

882550 

17.8 

i  n  Q 

927915 

42.  7 

072085 

64635 

76304 

44 

17 

810614 

24.  o 

882443 

1  r.C! 

no 

928171 

4O  7 

071829 

64657 

76286 

43 

18 

810763 

9A°8 

882336 

.0 

928427 

4O7 

071573 

64679 

76267 

42 

19 

810912 

24.  o 

882229 

i  '  .y 

928683 

Ao  fj 

071317 

64701 

76248 

41 

20 

811061 

OA  8 

882121 

17.9 

1  •*•   <  1 

928940 

4;$  .  / 

4O  *7 

071060 

64723 

76229 

40 

21 

9.811210 

OS 

9.882014 

17.9 

9.929196 

Z.I 

10.070804 

64746 

76210 

39 

22 

811358 

c\   ft 

881907 

17.9 

929452 

AO 

070548 

64768 

76192 

38 

23 

24 

811507 
811655 

24  ' 

24.7 

O  \  7 

881799 
881692 

17.9 
17.9 

929708 
929964 

42!  7 

070292 
070036' 

64790 

64812 

76173 
76154 

37 
36 

25 

811804 

24.  i 

O  4   fj 

881584 

17.9 

930220 

42.  6 

069780 

64834 

76135 

35 

26 

27 

811952 
812100 

24.  i 
24.7 

O  A   **1 

881477 
881369 

17.9 
17.9 

930475 
930731 

42.6 
42.6 

069525 
069269 

64856 

64878 

76116 
76097 

34 
33 

28 

812248 

24.  i 

881261 

17.9 

930987 

42.6 

069013 

64901 

76078 

32 

29 

812396 

24.7 

881153 

18.0 

931243 

42.6 

068757 

64923 

76059 

31 

30 

812544 

24.6 

881046 

18.0 

1  Q  A 

931499 

42.6 

068501 

64945 

76041 

30 

31 

9.812692 

OAfi 

9.880938 

lo  .  U 

1  Q  A 

9.931755 

42  .  6 

10.068245 

64967 

76022 

29 

32 
33 
34 

812840 
812988 
813135 

24> 
24.6 

OH  ft 

880830 
880722 
880613 

lo  .  0 
18.0 
18.0 

1  Q  A 

932010 
932266 
.  932522 

42!e 

42.6 

067990 
067734 
067478 

64989 
65011 
65033 

76003 
75984 
75965 

28 
27 
26 

35 

813283 

24.  0 

880505 

lo  .  0 

932778 

42  .  6 

067222 

65055 

75946 

25 

36 

813430 

C\A'  K. 

8o<3397 

18.0 

933033 

42.6 

066967 

65077 

75927 

24 

37 

813578 

24.  5 

880289 

18.0 

933289 

42.  o 

066711 

65100 

75908 

23 

38 

813725 

OA  K 

880180 

18.  1 

1  Q  1 

933545 

42.  o 

066455 

65122 

75889 

22 

39 

813872 

24.  O 

OH   C 

880072 

lo  .  1 

933800 

42.O 

066200 

65144 

75H70 

21 

40 

814019 

24.5 

879963 

18. 

934056 

42.6 

065944 

65166 

75851 

20 

41 

9.814166 

OA  K. 

9.879855 

18. 

1  Q 

9.934311 

42.6 

10.065689 

65188 

75832 

19 

42 

814313 

24,  O 

879746 

lo  . 

934567 

4%w.D 

065433 

65210 

75813 

18 

43 

814460 

24.5 

879637 

18. 

934823 

42.6 

065177 

65232 

75794 

17 

44 

814607 

24,4 

OH   H 

879529 

18. 

935078 

42.6 

064922 

65254 

75775 

16 

45 

814753 

24.  4 

O  .4   A 

879420 

18. 

935333 

42.6 

034667 

65276 

75556 

15 

46 

814900 

24.4 

OH   H 

879311 

18. 

935589 

42.6 

4O  ft 

064411 

65298 

75738 

14 

47 

815046 

24,4 

879202 

18. 

935844 

42  .  o 
AC)  £; 

064156 

65320 

75719 

13 

48 

815193 

A   A 

879093 

18.2 

936100 

42  .  0 

063900 

65342 

75700 

12 

49 

815339 

24.4 

878984 

18.2 

936355 

42.6 

063645 

65364 

75680 

11 

50 

815485 

24.4 

878875 

18.2 

936610 

42.6 

063390  1  165386 

75661 

10 

51 

9.815631 

24.3 

OA  Q 

9.878766 

18.2 

•10  O 

9.936866 

42.6 

40  x 

10.063134 

65408 

75642 

9 

52 

815778 

24.  o 

878656 

lo  .  2 

937121 

iJ.o 

062879 

65430 

75623 

8 

53 

54 

815924 
816069 

24.3 
24.3 

878547 
878438 

18.2 
18.2 

937376 
937632 

42.5 
42.5 

062624 
062368 

65452 
65474 

75604 

75585 

7 
6 

55 
56 

816215 
816361 

24.3 
24.3 

878328 
878219 

18.2 
18.2 

937887 
938142 

42.6 
42.5 

062113 
061858 

65496 
!  65518 

75566 
75547 

5 
4 

57 

816507 

24.  3 

OH.  O 

878109 

18.3 

938398 

42  .  5 

061602  !i  65540 

75528 

3 

58 

816652 

24.  2 

877999 

18.3 

938653 

42.5 

061347^65562 

75509 

2 

59 

816798 

24.  2 

877890 

18.3 

938908 

42.5 

061092  05584 

75490 

1 

60 

816943 

24.2 

877780 

18.3 

939163 

42.5 

060837  i:65uU6 

7547  I 

0 

Cosine. 

Sine. 

Cotang. 

Tang.   !  N.  cos. 

X.siue. 

~ 

49  Degrees. 

62          Log.  Sines  and  Tangents.  (41°)  Natural  Sines.     TABLE  LT. 

' 

Sine. 

D.  10 

Cosine. 

D.  10 

Tang. 

D.  10 

Cotang.  i  N.  sine 

X.  cos 

0 

9.816943 

9.877780 

100 

9.939163 

10.0(50837 

65606 

75471 

60 

1 

817088 

Of 

877670 

lo  .  ^ 

10    Q 

939418 

4°  n 

060582 

6562fc 

75452 

59 

2 

817233 

*4*- 

877560 

lo  .  L 

939673 

~t*  .  O 
10  K 

000327 

65650 

75433 

58 

3 

817379 

24.2 

877450 

18.  c 

939928 

-tZ  .  O 

060072 

1  65672 

75414 

57 

4 

817524 

24.2 

877340 

18.  c 

940183 

42.5 

>1O  & 

059817 

!  65694 

75395 

56 

6 
6 

817668 
817813 

24.1 
24.1 

877230 
877120 

18.  J 
18.4 

940438 
940694 

4J.  O 
42.6 
40  K 

059562 
059306 

!  657  16 
65738 

75375 
75356 

55 
54 

7 

817958 

24.1 

877010 

18.4 

940349 

Z  .0 
40  rx 

059061 

65759 

75337 

53 

8 

818103 

24.  1 

876S99 

18.4 

941204 

Z  .  o 
to  ^ 

058796 

!  05781 

75318 

52 

9 

818247 

24.  1 

876789 

18.4 

941458 

•iZ  .  O 

058542 

:  65803 

75299 

51 

10 

818392 

24.1 

876678 

18.4 

941714 

42  .  c 

058286 

65825 

75280 

50 

11 

9.818536 

24.  1 

9.876568 

18.4 

9.941968 

/ID  K 

10.058032 

65847 

75261 

49 

12 

818681 

24.  C 

876457 

18.4 

942223 

4vJ  .O 
49  ^ 

057777 

!  65869 

75241 

48 

13 

818825 

24.1 

876347 

18.4 

942478 

057522 

65891 

75222 

47 

14 

818969 

24.  C 

876236 

18.4 

942733 

42  .5 

Act  f 

057267 

65913 

75203 

46 

15 

819113 

24.  C 

876125 

18.5 

942988 

'vZ  .  o 

057012 

65935 

75184 

45 

16 

819257 

24.  C 

876014 

18.5 

943243 

42  .  5 

056757 

65956 

75165 

44 

17 

819401 

24.0 

875904 

18.5 

943498 

42.5 

056502 

65978 

75146 

43 

18 

819545 

24.0 

875793 

18.5 

943752 

42.5 

056248 

66000 

75126 

42 

19 

819689 

23.9 

875682 

18.5 

944007 

42  .5 

055993 

66022 

76107 

41 

20 
21 

819832 
9.819976 

23.9 
23.9 

875571 

9.875469 

18.6 
18.6 

944262 
9.944517 

42.5 
42.5 

055738 
10.055483 

66044 
!  66066 

75088 
75069 

40 
39 

22 
23 

820120 
820263 

23.9 
23.9 

875348 
875237 

18.5 
18.5 

944771 
945026 

42  .5 
42.4 

055229 
054974 

66088 
166109 

75050 
75030 

38 
37 

24 

820403 

23.9 

875126 

18.5 

10  a 

945281 

42.4 

054719 

66131 

75011 

36 

25 

820550 

23.9 

875014 

18.  o 

945535 

44 

054465 

166153 

74992 

35 

26 

820693 

23.8 

874903 

18.6 

945790 

44 

054210 

1  66175 

74973 

34 

27 

820836 

23.8 

874791 

18.6 

946045 

Act  A 

053955 

|  66197 

74953 

33 

28 

8-20979 

23.8 

874680 

18.6 

946299 

4ii.4 

053701 

66218 

74934 

32 

29 

821122 

23.8 

874568 

18.6 

946554 

42.4 

053446 

!  66240 

74915 

31 

30 

821265 

23.8 

874456 

18.6 

946808 

42  .4 

053192  166262 

74896 

30 

31 
32 

9.821407 
821550 

23.8 
23.8 

J.  874344 
874232 

18.6 
18.6 

9.947063 
947318 

42.4 
42.4 

Act  A 

10.052937 
052682 

66284 
66306 

74876 

74857 

29 

28 

33 

821693 

23.8 

OQ  7 

874121 

18.7 

1  R  7 

947572 

412.  4 
42  4 

052428  66327 

74838 

27 

34 

821835 

Jo.  / 

874009 

J.O  .  1 

947826 

052174  166349 

74818 

26 

35 

821977 

23.7 

873896 

18.7 

948081 

42.  4 

051919  66371 

74799 

25 

36 

822120 

23.7 

873784 

18.7 

948336 

42.4 

051664  OO.S93 

74780 

24 

37 

822363 

23.7 

873672 

18.7 

948590 

42.4 

051410  ''•  66414 

74760 

23 

38 
39 

822404 
82-2546 

23.7 
23.7 

873560 
873448 

18.7 
18.7 

948844 
949099 

42.4 
42.4 

05  1  1  56  j  166436 
050901  166458 

74741 
74722 

22 
21 

40 

822688 

23.7 

OQ  1* 

873335 

18.7 

10  ry 

949353 

42.4 
42  4 

050647|  166480 

74703 

20 

41 

9.822830 

Jo  .  D 

9.873223 

lo  .7 

9.949607 

1  0.  050393  i1  66501 

-4683 

19 

42 

822972 

23.6 

873110 

18.7 

949862 

42.4 

050138  !  66523 

-4663 

18 

43 

823114 

23.6 

872998 

L8.8 

950116 

42.4 

049884  166545 

-4644 

17 

44 

823255 

23.6 

872885 

18.8 

950370 

42  .  4 

049630  66566 

*4625 

16 

45 

823397 

23.6 

872772 

L8.8 

950625 

42.  4 

049375  6658.S 

"4600 

15 

46 

823539 

23.6 

872659 

18.8 

950879 

42.  4 

049121  :  66610 

-4586 

14 

47 

823680 

23.6 

872547 

L8.8 

951133 

4J.4 

048867  66632 

4567 

13 

48 

823821 

23.5 

872434 

[8.8 

951388 

42.4 

0486121:66663 

4548 

12 

49 

823963 

23.5 

872321 

18.8 

951642 

42.4 

048358'  66675 

45-2-2 

11 

50 

824104 

23.5 

872208 

18.8 

951896 

42.4 

048104  66697 

74509 

10 

51 
52 

9.824245 
824386 

23.6 
23.5 

9.872095 
871981 

18.8 
18.9 

J.  952  150 
952405 

42.4 
42.4 

0.047850  66718 
047595  66740 

74489 
74470 

9 

8 

53 

824527 

23.5 

871868 

[8.9 

952659 

42.4 

047341  66762 

74451 

7 

54 

824668 

23.5 

871755 

[8.9 

95-2913 

12.4 

047087'  66783 

74431 

6 

55 

824808 

23.4 

871641 

[8.9 

953167 

t2  .  4 

046833  66805 

74412 

5 

56 

824949 

23.4 

871528 

[8.9 
8  9 

953421 

L2.  3 
[2  3 

046579|  60827 

74392 

4 

57 

826090 

~o  .  4 

871414 

953675 

046325  66848 

74373 

3 

58 
59 

825230 
825371 

23.4 
23.4 

871301 
871187 

.8.9 
18.9 

8f\ 

953929 
954183 

12.  3 
42.3 

[O  Q 

046071  i  66870 
045817  66891 

74353 
?4334 

2 
1 

60 

825511 

23  .4 

871073 

.  y 

954437 

t  Z  .  o 

045563  |  66913 

74314 

0 

Oosine. 

Sine. 

Cotang. 

Tang.   i  N.  cos. 

V.sine. 

' 

48  Degrees. 

TABLE  11.     Log.  Siiies  and  Tangents.  (42°)  Natural  Sines.          63 

' 

Sine. 

D.  10'' 

Cosine. 

I).  10"|   Tung. 

D.  10" 

Cotang. 

N.  sine. 

N.  cos. 

o 

9.825511 

9.871073 

9.954437 

10.045563 

66913 

74314 

60 

1 

825651 

23.4 

870960 

19.0 

954691 

42.3 

045309 

66935 

74295 

59 

2 

825791 

23.3 

870846 

19.0 

954915 

42.3 

/1O  «} 

045055 

66956 

74276 

58 

3 

825931 

23.3 

870732 

19.0 

955200 

4J  .0 

044800 

66978  74256 

57 

4 

826071 

23.3 

870618 

19.0 

955454 

42.3 

044646  166999 

74237 

56 

6 

826211 

23.3 

870504 

19.0 

955707 

42.3 

044293  167021 

74217 

55 

6 

7 

826351 
826491 

23.3 
23.3 

870390 
870276 

19.0 
19.0 

955961 
956215 

42.3 
42.3 

044039 
043785 

67043  74198 
67064174178 

54 
53 

8 

826631 

23.3 

870161 

19.0 

95(5469 

42.3 

043531 

67086 

74159 

62 

9 

826770 

23.3 

oo  o 

870047 

19.0 

956723 

42.3 

ACt  O 

043277 

67107 

74139 

51 

10 

826910 

2o  .2 

8li9933 

19. 

956977 

•1_  .0 

043023 

67129 

74120 

50 

11 

9.827049 

23.2 

9.869818 

19. 

9.957231 

42.3 

10.042769 

67151 

74100 

49 

19 

827189 

23.2 

869704 

19. 

957485 

42.3 

042515 

67172 

74080 

48 

13 

827328 

23.2 

869589 

19. 

957739 

42.3 

042261 

67194 

74061 

47 

14 

827467 

23  .  2 

869474 

19. 

957993 

42.3 

042007 

67215 

74041 

4(5 

15 

827606 

23.2 

869360 

19. 

958246 

42.3 

041754 

67237 

74022 

45 

1G 

827745 

23.2 

869245 

19. 

958500 

42.3 

041500 

67268 

74002 

44 

17 

827884 

23.2 

869130 

19. 

958754 

42.3 

041246 

i  67280 

73983 

43 

18 

828023 

23.1 

869015 

19. 

959008 

42-3 

040992 

67301 

73963 

42 

19 

828162 

23.  1 

868900 

19.2 

959262 

42.3 

040738 

87323 

73944 

41 

20 

828301 

23.1 

868785 

19.2 

959516 

42.3 

040484 

67344 

73924 

40 

21 

9.828439 

23.1 

9.868670 

19.2 

9.959769 

42.3 

10.040231 

67366 

73904 

39 

22 

828578 

23.1 

868555 

19.2 

960023 

42.3 

039977 

67387 

73885 

38 

23 
24 

828716 

828855 

23  .  1 
23.1 

868440 
86H324 

19.2 
19.2 

960277 
960531 

42.3 
42.3 

039723 
039469 

67409 
67430 

73865 
73846 

37 
36 

25 

828993 

23.0 

868209 

19.2 

960784 

42.3 

039216 

67452 

73826 

35  i 

26 

829131 

23.0 

868093 

19.2 

961038 

42.3 

038962 

67473 

73806 

34 

27 

829269 

23.0 

867978 

19.2 

961291 

42.3 

038709 

(57495 

73787 

33 

28 

829407 

23.0 

867862 

19.3 

961545 

42.3 

038455 

67516 

73767 

32 

29 

829545 

23.0 

867747 

19.3 

961799 

42.3 

038201  67538 

73747 

31 

30 

829683 

23.0 

867631 

19.3 

962052 

42.3 

037948 

67559 

73728 

30 

31 
32 

9.829821 
829959 

23.0 
22.9 

9.867515 
867399 

19.3 
19.3 

9.962306 
962560 

42.3 
42.3 

10.037694 

037410 

67580 
1  67602 

73708 

73688 

29 
28 

33 

830097 

22.9 

OO  O 

867283 

19.3 

962813 

42.3 

037187  1|  67623 

73(5(59 

27 

34 

830234 

ZA  -9 

867167 

19.3 

963067 

42  .3 

036933  j  67645 

73(549 

26 

35 
3G 

830372 
830509 

22.9 
22.9 
90  q 

867051 
866935 

19.3 
19.3 
in  A 

963320 
963574 

42.3 
42.3 

AC)   O 

036(580 
03(542(5 

67666 

67688 

TIJG-'iJ 
73(510 

25 
24 

37 
38 

830646 
830784 

£<&  •  d 
22.9 

OO  Q 

866819 
866703 

LJ  ,  4 
19.4 
in  A 

963827 
964081 

*t^t  .  O 

42.3 

4O  Q 

036173 
035919 

67709(73590 
6773073570 

23 

22 

39 

830921 

££  .  y 

866586 

Iv  .4 

964335 

.w  .  O 

035665 

6775273551 

21 

40 

831058 

22.8 

866470 

19.4 

964588 

42.3 

035412 

6777373531 

20 

41 

9.831195 

22.8 

9.866353 

19.4 

9.964842 

42.2 

10.035158 

67795)73511 

19 

42 

831332 

22.8 

no  Q 

866237 

19.4 

965095 

42.2 

034905 

6781673491 

18 

43 

831469 

22  .0 

866120 

19.4 

965349 

42  .2 

034651 

67837173472 

17 

44 

831606  lg-2 

866004 

19.4 

965602 

42.2 

034398 

67859 

73452 

16 

45 

831742i^-g 

865887 

19.5 

965855 

42-2 

034145 

678HO 

73432 

15 

46 

47 

48 

831879 
832015 
832152 

1  Z^i  -O 

22.8 
22.7 

865770 
865653 
865536 

19.  B 
19.5 
19.5 

966109 
966362 
966616 

42.2 
42.2 
42.2 

033891 

033638 
033384 

67901 
67923 

67944 

73413 
73393 
73373 

14 
13 

12 

49 

832288  ™'' 

865419 

19.5 

966869 

42.2 

033131 

67965 

73353 

11 

50 

832425  —  •' 

865302 

19.5 

967123 

42.2 

032877 

67987 

73333 

10 

51 

9  883661  g. 

9.865185 

19.5 

9.967376 

42.2 

10.032624 

68008 

73314 

9 

52 

883697  Sri 

865068 

19.5 

967629 

42  .2 

032371 

68029 

73294 

8 

53 

832833  '"*•' 

864950 

19.5 

9(57883 

42.2 

032117 

68051 

73274 

7 

54 

882969  jfj'fi 

864833 

19.5 

968136 

42.2 

031864 

68072 

73254 

6 

55 

833105  g'J 

864716 

19.6 

968389 

42.2 

031611 

68093 

73234 

5 

56 

833241  ^ 

864598 

19.6 

968(543 

42.2 

031357 

68115 

73215 

4 

57 

833377  g'J 

864181 

19.6 

968896 

42.2 

031104 

(5813(5 

73195 

3 

58 

833512  g-J 

864363 

19.6 

969149 

42.2 

030851  <;81.:i*< 

7:3175 

<2 

59 

833648  :~~7 

864245 

19.6 

969403 

42.2 

030597  (581  7!) 

73155 

T 

60 

833783  r^'b 

864127 

19.6 

969656 

42.2 

030344'  68200 

/3136 

0 

Cosine. 

Bine. 

Cotang. 

Tang.   H  N.  cos 

N.sine 

' 

47  Degrees. 

Log.  Sines  and  Tangents.  (43°)  Natural  Sines.     TABLE  II. 

' 

Sine. 

D.  10 

Cosine. 

D.  10' 

Tang.   |D.  10"   C.itsmg.  jj.N.sine. 

N.  cos 

o 

9.833783 

oo  r 

9.864127 

9.969656 

40  o 

10.030344 

1  68200 

73135 

60 

] 

833919 

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09  r 

864010 

1  (\   p 

969909 

z  .  2 
40  o 

030091 

68221 

73  IK 

59 

IT 

834054 
834189 

22!  5 

OO  K 

863892 
863774 

1LJ  .  u 

19.7 

970162 
970416 

2  .  2 

42.2 

029838 
029584 

68242 
!  68264 

73096 
73076 

58 
57 

i 

834325 

22,  5 

99  t 

863656 

19.7 

1O  T 

970669 

42.2 

/io  o 

029331  68285 

73056 

56 

5 

834460 

~  w  .  £ 

OO  K. 

863538 

iy  .  7 

970922 

4^.  2 

029078 

68306 

73036 

55 

6 

834595 

<**>  .  £ 
99  - 

863419 

in  T 

971175 

AVr 

028825 

68327 

73016 

54 

rt 

834730 

<6-6.  D 

OO  K 

863301 

iy  .  7 

in  T 

971429 

40  o 

028571 

68349 

72991 

53 

8 

834865 

*'&.O 

863183 

iy  .  7 

971682 

2.2 

028318 

68370 

7297( 

52 

9 

834999 

22  4 

863064 

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971935 

42  2 

028065 

68391 

72957 

51 

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835134 

862946 

iy  .  t 

i  q  o 

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027812 

68412 

72937 

50 

11 

9.835269 

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9.862827 

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40  O 

10.027559 

68434 

72917 

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835403 

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862709 

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in  Q 

972694 

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027306 

68455 

72897 

48 

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835538 

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862590 

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972948 

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027052 

68476 

72877 

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14 

835672 

22.4 

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862471 

19.  £ 

1  Q   t 

973201 

42,2 

/1O  O 

026799 

68497 

72857 

46 

15 

835807 

2&  .  4 

09  A 

862353 

iy  .  c 
19  S 

973454 

4*5  ,  2 

42  2 

026546 

68518 

72837 

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16 

835941 

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862234 

973707 

026293 

68539 

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836075 

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862115 

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973960 

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026040 

68561 

72797 

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836209 

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861996 

iy  .0 

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974213 

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025787 

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72777 

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836477 

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22.3 

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861877 
861758 

iy  .  o 
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974466 
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831519 

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975226 

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024774 

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861400 

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024521 

68688 

72677 

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24 

837012 

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861280 

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975732 

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024268 

68709 

72657 

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25 

837146 

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861161 

iy  .  y 

in  n 

975985 

42  .  2 

024015 

68730|72637 

35 

26 

837279 

22  .  2 
99  9 

861041 

iy  .  y 

i  q  q 

976238 

42  2 

023762 

68751 

72617 

34 

27 

837412 

—  -  .  M 

99  9 

860922 

iy  .  y 
1  q  q 

976491 

/IO  O 

023509 

68772 

72597 

33 

28 

837546 

22  .  £ 
99  9 

860802 

iy  .  y 
IQ  Q 

976744 

4-^  .  2 

023256 

68793 

72577 

32 

29 
30 
31 

837679 
837812 
9.837945 

22.  A 

22.2 
22.2 

860682 
860562 
9.860442 

iy  .  y 
20.0 
20.0 

976997 
977250 
9.977503 

42^2 
42.2 

023003 
022/50 
10  022497 

68814 
68835 
68857 

72557 
72537 
72517 

31 

30 
29 

32 

838078 

22.2 

OO  1 

860322 

20  .  0 
on  n 

977756 

42.2 

022244 

68878 

72497 

28 

33 

838211 

22  .  1 

860202 

•*U  .  U 

978009 

42.2 

021991 

68899 

72477 

27 

34 
35 

838344 
838477 

22.  1 
22.1 

OO  1 

860082 
859962 

20.0 

20.0 
20  0 

978262 
978515 

42.2 
42.2 
42  2 

021738 
021485 

6892072457 
6894172437 

26 
25 

36 

838610 

w  W  .  1 

OO  1 

859842 

978768 

021232 

68962 

72417 

24 

37 

838742 

2&  •  1 

859721 

20.0 

979021 

42.2 

0-20979 

68983 

72397 

23 

38 
39 

838875 
839007 

22.  1 
22.1 

OO   1 

859601 
859480 

20!  1 

•>M  1 

979274 
979527 

42.2 

020726 
020473 

69004 

69025 

72377 

72357 

22 
21 

40 

839140 

22  .  1 

859360 

<£(J  .  1 
on  i 

979780 

4^.2 

020220 

69046;72337 

20 

41 

.839272 

22.0 

99  n 

859239 

20  .  1 
on  i 

9.980033 

42.2 
42  ^ 

0.019967 

69067(72317 

19 

42 

839404 

-  -  .  U 

859119 

•*U.  1 

980286 

019714 

6908872297 

18 

43 

839536 

22.0 

858998 

20.  1 

On  1 

980538 

42.2 

019462 

69109  72277 

17 

44 

839668 

22.  0 

858877 

2\J  .  1 

930791 

42.  2 

019209 

6913072257 

16 

45 

839800 

22.0 

858756 

n   n 

981044 

4~.  1 

018956 

69151  72236 

15 

46 

839932 

22.0 

858635 

20  ,  "2 
on  o 

981297 

42.  1 

AC)   I 

018703 

6917272216 

14 

47 

48 
49 

840064 
840196 
840328 

22.  0 
21.9 
21.9 

858514 
858393 
858272 

20J2 
20.2 

981550 
981803 
982056 

42  .  i 
42.1 
42.1 

018450 
0181971 
017944 

69193  72196 
69214172176 
69235  !72  156 

13 
12 
11 

50 

840459 

21.9 

858151 

20.  2 

982309 

42.  1 

017691 

6925672136 

10 

51 

.840591 

21.9 

.858029 

20.  2 

.982562 

1:2.  1 

0.0174381 

6927772116 

9 

52 

840722 

21.9 

857903 

20.2 

982814  1  ^'f 

01  7186  ;  6929872095 

8 

53 

840854 

21.9 

857786 

2J.2 

983067 

016933:16931972075 

7 

54 

840985 

21  .9 

857665 

20.2 

933320 

42.  1 

016U80  69340172055 

6 

55 
56 

841116 

841247 

21  .9 

21.8 

857543 
85-422 

20.  3 
20.3 

983573 
983826 

42.  1 
42.1 

016427;j  69361  !72035 
016174'  69382^72015 

5 
4 

57 

841378 

21  .8 

85730J 

20.3 

934079 

42.  1 

015921  69403:71995 

3 

58 
59 

841509 
841640 

21  .8 
21.8 

85  71  78  | 

20.3 
20.3 

984331 
984584 

42.  1 
42.1 

015669  694-2471974 
015416  b9445  71954 

2 
1 

841771 

21.8 

85J934 

20.3 

984837 

42.  1 

015163  69466  j  7  1934 

0 

('..sine. 

Sine. 

Cotang. 

Tang.   :  N.  cos.  N.siu<;. 

' 

46  Degrees. 

TABLE  II.     Log.  Sines  and  Tangents.  (44°)  Natural  Sines.          65 

' 

Sine. 

D.  10" 

Cosine. 

D.  10" 

Tang. 

D.,10" 

Cotang. 

N.  sine. 

N.  cos. 

0 

9.841771 

9.856934 

9.984837 

10.015163 

69466 

71934 

60 

1 

841902 

21.8 

856812 

20.3 

985090 

42. 

014910 

69487 

71914 

59 

2 

842033 

21.8 

856690 

20.3 

985343 

42. 

014657 

69508 

71894 

58 

3 

842163 

21.8 

856568 

20.4 

985596 

42. 

014404 

69529 

71873 

57 

4 

842294 

21.7 

856446 

20.4 

985848 

42. 

014152 

69549 

71853 

56 

5 

842424 

21.7 

856323 

20.4 

986101 

42. 

013899 

69570 

71833 

55 

6 

842555 

21.7 

856201 

20.4 

986354 

42. 

013646 

69591 

71813 

54 

7 

842G85 

21.7 

856078 

20.4 

986607 

42.1 

013393 

69612 

71792 

53 

8 

842815 

21.7 

855956 

20.4 

986860 

42.1 

013140 

69633 

71772 

52 

9 

842946 

21.7 

855833 

20.4 

987112 

42.  1 

012888 

69654 

71752 

51 

10 

843076 

21.7 

855711 

20.4 

987365 

42.1 

012635 

69675 

71732 

50 

11 

9.843206 

21.7 

9.855588 

20.5 

9.987618 

42.  1 

10.012382 

69696 

71711 

49 

12 

843336 

21.6 

O1   £* 

855465 

20.5 
90  Ft 

987871 

42.1 
49  1 

012129 

69717 

71691 

48 

13 

843466 

21.  b 

855342 

**\J  .  O 
90  fi 

988123 

Q-6  .  1 

49  1 

011877 

69737 

71671 

47 

14 

843595 

21  .6 

O1   £? 

855219 

^U.  O 
90  K 

988376 

4^.  1 
49  1 

011624 

69758 

71650 

46 

15 

843725 

yi.b 

855096 

A\J  .  0 
9O  f\ 

988629 

4^.  1 

49 

011371 

69779 

71630 

45 

16 

843855 

21.6 

854973 

^\J  .  0 

on  K 

988882 

4^. 
49 

011118 

69800 

71610 

44 

17 

843984 

21.6 

854850 

4\J  .  0 

989134 

*±£  . 

ACt 

010866 

69821 

71590 

43 

18 

844114 

21.6 

854727 

20.5 

989387 

42  . 

010613 

69842 

71569 

42 

19 

844243 

21.5 

854603 

20.6 

989640 

42. 

010360 

69862 

71549 

41 

20 

844372 

21.5 

854480 

20.6 

989893 

42. 

010107 

69883 

71529 

40 

21 

9.844502 

21.5 

9.854356 

20.6 

9.990145 

42. 

10.009855 

69904 

71508 

39 

22 

844631 

21.5 

854233 

20.6 

990398 

42.1 

009602 

69925 

71488 

38 

23 

844760 

21.5 

854109 

20.6 

990651 

42.1 

009349 

69946 

71468 

37 

24 

844889 

21.5 

853986 

20.6 

990903 

42.1 

009097 

69966 

71447 

36 

25 

845018 

21.5 

853862 

20.6 

991156 

42.1 

008844 

69987 

71427 

35 

26 

845147 

21.5 

853738 

20.6 

991409 

42.1 

008591 

70008 

71407 

34 

27 

845276 

21.5 

853614 

20.6 

991662 

42.1 

ACt  1 

008338 

70029 

71386 

33 

28 

845405 

21.4 

853490 

20.  7 

991914 

42.1 

in  . 

008086 

70049 

71366 

32 

29 

845533 

21.4 

853366 

20.7 

992167 

42.1 

ACt  1 

007833 

70070 

71345 

31 

30 

845662 

21.4 

853242 

20.7 

992420 

42.1 

OQ7580 

70091 

71325 

30 

31 

9.845790 

21.4 

9.853118 

20.7 

9.992672 

42. 

10-007328 

70112 

71305 

29 

3-2 

845919 

21.4 

852994 

20.7 

992925 

42. 

007075 

70132 

71284 

28 

33 

846047 

21.4 

852869 

20.7 

993178 

42. 

006822 

70153 

71264 

27 

34 

846175 

21.4 

852745 

20.7 

993430 

42. 

006570 

70174 

71243 

26 

35 

846304 

21.4 

852620 

20.7 

993683 

42. 

006317 

70195 

71223 

25 

36 

846432 

21.4 

852496 

20.7 

993936 

42. 

006064 

70215 

71203 

24 

37 

846560 

21.3 

852371 

20.8 

994189 

42. 

005811 

70236 

71182 

23 

38 

846688 

21.3 

852247 

20.8 

994441 

42. 

005559 

70257 

71162 

22 

39 

846816 

21.3 

852122 

20.8 

994694 

42. 

005306 

70277 

71141 

21 

40 

846944 

21.3 

851997 

20.8 

994947 

42. 

005053 

70298 

71121 

20 

41 

9.847071 

21.3 

9.851872 

20.8 

9.995199 

42. 

10-004801 

70319 

71100 

19 

42 

847199 

21.3 

851747 

20.8 

995452 

42. 

004548 

70339 

71080 

18 

43 

847327 

21.3 

851622 

20.8 

995705 

42.  1 

004295 

70360 

71059 

17 

44 

847454 

21.3 

851497 

20.8 

995957 

42.1 

004043 

70381 

71039 

16 

45 

847582 

21.2 

851372 

20.9 

996210 

42.1 

003790 

70401 

71019 

15 

46 

847709 

21.2 

851246 

20.9 

996463 

42.1 

003537 

70422 

70998 

14 

47 

847836 

21.2 

851121 

20.9 

996715 

42. 

ACt 

003285 

70443 

70978 

13 

48 

847964 

21.2 

850996 

20.9 

996968 

4.4. 

ACt 

003032 

70463 

70957 

12 

49 

848091 

21.2 

850870 

20.9 

997221 

<±Z  . 

ACt 

002779 

70484 

70937 

11 

50 

848218 

21.2 

850745 

20.9 

997473 

Q"  . 

ACt 

002527 

70505 

70916 

10 

-51 

9.848345 

21.2 

9.850619 

20.9 

9.997726 

41*  . 

ACt 

10.002274 

70525 

70896 

9 

52 

848472 

21.2 

850493 

20.9 

997979 

4». 

ACt 

002021 

70546 

70875 

8 

'  53 

848599 

21.1 

850368 

21.0 

998231 

4z. 

001769 

70567 

70855 

7 

54 

848726 

21.1 

850242 

21.0 

998484 

42. 

001516 

70587 

70834 

6 

55 

848852 

21.1 

850116 

21.0 

998737 

42. 

001263 

70608 

70813 

5 

56 

848979 

21.1 

849990 

21.0 

998989 

42. 

001011 

70628 

70793 

4 

57 

849106 

21.1 

849864 

21  .0 

999242 

42. 

ACt 

000758 

70649 

70772 

3 

58 

849232 

21.  1 

849738 

21.0 

999495 

*». 

ACt 

000505 

70670 

70752 

2 

59 

849359 

21.  1 

849611 

21  .0 

999748 

QX>  . 

ACt 

000253 

70690 

70731 

1 

60 

849485 

21.1 

849485 

21.0 

10.000000 

4-i. 

000000 

70711 

70711 

0 

Cosine. 

Sine" 

Co  tang. 

Tang. 

N.  cos. 

N.pinc. 

' 

45  Degrees. 

66           LOGARITHMS 

TABLE  III. 

LOGARITHMS  OF  NUMBERS. 

FROM  1  TO  200, 

INCLUDING  TWELVE  DECIMAL  PLACES. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 
2 
3 

4 
5 

000000  000000 
301029  995664 
477121  254720 
602059  991328 
698970  004336 

41 
42 
43 
44 
45 

612783  856720 
623249  290398 
633468  455580 
643452  676486 
653212  513775 

81 

82 
83 
84 
85 

908485  018879 
913813  852384 
919078  092376 
924279  286062 
929418  925714 

6 
7 
8 
9 
10 

778151  250384 
845098  040014 
903089  986992 
954242  509439 
Same  as  to  1. 

46 

47 
48 
49 
50 

662757  831682 
672097  857926 
681241  237376 
690196  080028 
Same  as  to  5. 

86 
87 
88 
89 
90 

934498  451244 
939519  252619 
944482  672150 
949390  006645 
Same  as  to  9. 

11 
12 
13 
14 
15 

041392  685158 
079181  246048 
113943  352307 
146128  035678 
176091  259056 

51 
52 
53 
54 
55 

707570  176098 
716003  343635 
724275  869601 
732393  759823 
740362  689494 

91 
92 
93 
94 
95 

959041  392321 
963787  827346 
968482  948554 
973127  853600 
977723  605889 

16 
17 
18 
19 
20 

204119  982656 
230448  921378 
255272  505103 
278753  600953 
Same  as  to  2. 

56 
57 
58 
59 
60 

748188  027005 
755874  855672 
763427  993563 
770852  011642 
Same  as  to  6. 

96 
97 
98 
99 
100 

982271  233040 
986771  734266 
991226  075692 
995635  194598 
Same  as  to  10, 

"  21 
22 
23 
24 
25 

322219  2947 
342422  680822 
361727  836018 
380211  241712 
397940  008672 

61 
62 
63 
64 
65 

785329  835011 
792391  699498 
799340  549453 
806179  973984 
812913  356643 

101 
102 
103 
104 
105 

004321  373783 
008600  171762 
012837  224705 
017033  339299 
021189  299070 

26 
27 
28 
29 
30 

414973  347971 
431363  764159 
447158  031342 
462397  997899 
Stnno  as  to  3. 

66 
67 
68 
69 
70 

819543  935542 
826074  802701 
832508  912706 
838849  090737 
Same  as  to  7. 

103 
107 
108 
109 
110 

025305  865265 
029383  777685 
033423  755487 
037426  497941 
Same  as  to  11. 

31 
32 
33 
34 
35 

491361  693834 
505149  978320 
518513  939878 
531478  917042 
544068  044350 

71 
72 
73 
74 
75 

851258  348719 
857332  496431 
863322  860120 
869231  719731 
875061  263392 

111 
112 
113 
114 
115 

045322  978787 
049218  022670 
053078  443483 
056904  851336 
060397  840354 

36 
37 
38 
39 
40 

556302  500767 
568-201  724067 
579783  596617 
591064  607026 
Same  as  to  4. 

76 
77 
78 
79 
80 

880813  592281 
886490  725172 
892094  602690 
897627  091290 
Same  as  to  8. 

116 
117 
118 
119 
120 

064457  989227 
068185  861746 
071882  007306 
075546  961393 
Same  as  to  12. 

OF  NUMBERS.           67 

N. 

Log. 

N. 

148 
149 
150 
151 

:  152 

Log. 

N.       Log 

121 
122 
123 
124 

125 

082785  370316 
086359  830675 
089905  111439 
093421  685162 
096910  013008 

170261  715395 
173186  268412 
176091  259056 
178976  947293 
181843  587945 

175 
176 
177 
178 
179 

243038  048686 
245512  667814 
247973  266362 
250420  002309 
252853  030980 

126 
127 
128 
129 
130 

100370  545118 
103803  720956 
10/209  969G48 
110589  710299 
Same  as  to  13. 

153 
154 
155 

i  156 
I  157 

184691  430818 
187520  720836 
190331  698170 
193124  588354 
195899  652409 

180 
181 

182 
183 

184 

255272  505103 
257678  574869 
260071  387985 
262451  089730 
264817  823010 

131 

132 
133 
134 

135 

117271  295656 
120573  931206 
123851  640967 
127104  798365 
130333  768495 

158 
159 
160 
161 
162 

198657  086954 
201397  124320 
204119  982656 
206825  876032 
209515  014543 

185 
186 
187 
188 
189 

267171  728403 
269512  944218 
271841  606536 
274157  849264 
276461  804173 

136 
137 
138 
139 
140 

133538  908370 
136720  567156 
139879  086401 
143014  800254 
146128  035678 

163 
164 
165 
166 
167 

212187  604404 
214843  848048 
217483  944214 
220108  088040 
222716  471148 

190 
191 
192 
193 
194 

278753  600953 
281033  367248 
283301  228704 
285557  309008 
287801  729930 

141 
142 
143 
144 
145 

149219  112655 
152288  344383 
155336  037465 
158362  492095 
161368  002235 

168 
169 
170 
171 
172 

225309  281726 
227886  704614 
230448  921378 
232996  110392 
235528  446908 

195 
196 
197 
198 
199 

290034  611362 
292256  071366 
294466  226162 
296665  190262 
298853  076410 

146 
147 

164352  855784 
167317  334748 

173 
174 

238046  103129 
240549  248283 

LOGARITHMS  OF  THE  PRIME  NUMBERS 

FROM  200  TO  1543, 

INCLUDING  TWELVE  DECIMAL  PLACES. 

N. 

Log. 

N.      Log. 

N. 

Log. 

201 
203 
207 
209 
211 

303196  057420 
307496  037913 
315970  345457 
320146  286111 
324282  455298 

277 
281 
283 
293 
307 

442479  769064 
448706  319905 
451786  435524 
466867  620354 
487138  375477 

379 
383 
389 
397 
401 

678639  209968 
583198  773968 
589949  601326 
598790  506763 
603144  372G20 

223 

227 
229 
233 
239 

348304  863048 
356025  857193 
359835  482340 
367355  921026 
378397  900948 

311 
313 
317 
331 

337 

492760  389027 
495544  337546 
501059  262218 
519827  993776 
527629  900871 

409 
419 
421 
431 
433 

611723  308007 
622214  022966 
624282  095836 
634477  270161 
636487  896353 

241 
251 
257 
263 
269 

3820T7  042575 
399673  721481 
409933  123331 
419955  748490 
429762  280002 

347 
349 
353 
359 
367 

540329  474791 
542825  426959 
647774  705388 
555094  448578  1 
564666  064252 

439 
443 
449 
457 
461 

642424  520242 
646403  726223 
652246  341003 
659916  200070 
663700  926390 

271 

432969  290874 

373 

571708  831809 

463 

666580  991018 

68           LOGARITHMS 

N. 

Lug. 

N. 

Log. 

N. 
TfrT 

1181 
1187 
1193 
1201 

Log. 

467 
479 
487 
491 
499 

6,)9olo  6805t,6 
680335  513414 
687528  961215 
691081  492123 
698100  545623 

821 
823 
827 
829 
839 

914343  157119 
915399  835212 
917505  509553 
918554  530550 
923761  960829 

ObS5c6  895072 
072249  807613 
074450  718955 
076640  443670 
OJ9543  007385 

503 
609 
521 

623 

541 

701567  985056 
706717  782337 
716837  723300 
718501  688867 
733197  285107 

853 
857 
859 
863 

877 

930949  031168 
932980  821923 
933993  163831 
936010  795715 
942099  593356 

1213 
1217 
1223 
1229 
1231 

083860  800845 
085290  678210 
087426  458017 
089551  882866 
090258  052912 

647 
657 
663 
669 
671 

737987  326333 
745855  195174 
750508  394851 
755112  266393 
756636  108246 

881 
883 
887 
907 
911 

944975  908412 
945960  703578 
947923  619832 
957607  287060 
959518  376973 

1237 
1249 
1259 
1277 
1279 

092369  699609 
096562  438356 
100025  729204 
108190  896808 
106870  642460 

577 
687 
593 
699 
601 

761176  813156 
768638  101248 
773054  693364 
777426  822389 
778874  472002 

919 
929 
937 
941 

947 

963315  611386 
988015  713994 
971739  590888 
973589  623427 
976349  979003 

1283 
1289 
1291 
1297 
1301 

108226  656362 
110252  917337 
110926  242517 
112939  986066 
114277  296540 

607 
613 
617 
619 
631 

783138  691075 
787460  474518 
790285  164033 
791690  649020 
800029  369244 

953 
967 
971 
977 
983 

979092  900638 
985426  474083 
987219  229908 
989894  563719 
992553  517832 

1303 
1307 
1319 
1321 
1327 

114944  415712 
116275  587564 
120244  795568 
120902  817604 
122870  922849 

641 
643 
'  647 
653 
659 

806858  029519 
808210  972924 
810904  280669 
814913  181275 
818885  414594 

991 
997 
1009 
1013 
1019 

996073  654485 
998695  158312 
003891  166237 
005609  445360 
008174  184006 

1361 
1367 
1373 
1381 
1399 

133858  125188 
135768  514554 
137670  537223 
140193  678544 
145817  714122 

661 
673 

677 
683 
691 

810201  459486 
828015  064224 
830588  668685 
8344-20  703682 
839478  047374 

1021 
1031 
1033 
1039 
1049 

0090-25  742087 
013258  665284 
014100  321520 
016615  547557 
020775  488194 

1409 
1423 
1427 
1429 
1433 

148910  994096 
153204  896557 
154424  012366 

155032  228774 
156246  402184 

701 
709 
719 
727 
733 

845718  017967 
850646  235183 
856728  890383 
861534  410859 
865103  974742 

1051 
1061 
1063 
1069 
1087 

021602  716028 
025715  383901 
026533  264523 
028977  705209 
036229  644086 

1439 
1447 
1451 
1453 
1459 

158060  793919 
160468  531109 
161667  412427 
162265  614286 
164055  291883 

739 
743 
751 
757 
761 

888644  488395 
870988  813761 
855639  937004 
879095  879500 
881384  656771 

1091 
1093 
1097 
1103 
1109 

037824  750588 
038620  161950 
040206  627575 
042595  512440 
044931  546  119 

1471 
1481 
1483 
1487 
1489 

167612  672629 
170555  058512 
171141  151014 
172310  968489 
172894  731332 

769 
773 
787 
797 
809 

885926  339801 
888179  493918 
895974  732359 
901458  32139G 
907948  521612 

1117 
1123 
1129 
1151 
1153 

018053  173116 
050379  756261 
052693  941925 
061075  323630 
061829  307295 

1493 
1499 
1511 
1523 
1531 

174059  807708 
175801  632866 
179264  464329 
182699  903324 
184975  190807 

811 

909020  854211 

1163 

065579  714728 

1543 

188365  926053 

OF    NUMBERS. 


69 


AUXILIARY    LOGARITHMS, 


N. 

l«g. 

W. 

Log. 

i.ouy 

003891106237  - 

1 

1.0009 

000390689248  ^ 

.008 

003460532110 

1  .  0008 

000347296684 

.007 

003029470554 

1  .  0007 

000303899784 

.006 

002598080685 

1.0006 

001)260498547 

.005 

002166061766 

A 

1  .  0005 

000217092970 

>B 

.004 

001733712775 

1  .  0004 

000173683057 

.003 

001300933020 

1  .  0003 

000130268804 

.002 

000867721529 

1  .  0002 

000086850211 

1.001 

000434077479 

1  .  0001 

000043427277 

N. 

Log. 

:   N. 

Log. 

.  00009 
.  00008 
.  00007 
.  00008 
.  00005 
.  00004 
.00003 
.  00002 
.  00001 

000039083266 
000034740691 
000030398072 
0000-26055410 
000021712704 
000017371430 
0000130-28638 
000008085802 
0001104342923 

1.000009 
.  000008 
.000007 
.000006 
.  000005 
.  000004 
1  .  000003 
1.000002 
1.000001 

OOOOJ3908628 
000003474338 
000003040047 
000002605756 
000002171464 
000001737173 
000001302880 
OU0000868587 
000000434294 

- 

N. 

Log. 

1.0000001 
1.00000001 
1.000000001 
1  .  0000000001 

000000043429  (n) 
000000001343  (o) 
000000000434  (p) 
000000000043  (q) 

?7i=0.4342944819        log.  —1.637784298. 

By  the  preceding  tables  —  and  the  auxiliaries  A,  B,  and 
C,  we  can  find  the  logarithm  of  any  number,  true  to  at  least 
ten  decimal  places. 

But  some  may  prefer  to  use  the  following  direct  formula, 
which  may  be  found  in  any  of  the  standard  works  on  algebra: 

Log.  (z-fl)==log.z+0.8685889638/'-_L>) 

The  result  will  be  true  to  twelve  decimal  places,  if  z  be 
over  2000. 

The  log.  of  composite  numbers  can  be  determined  by  the 
combination  of  logarithms,  already  in  the  table,  and  the  prime 
numbers  from  the  formula. 

Thus,  the  number  3083  is  a  prime  number,  find  its  loga 
rithm. 

We  first  find  the  log.  of  the  number  3082.  By  factoring, 
discover  that  this  is  the  product  of  46  into  67. 


70  NUMBERS. 


Log.  46,  1.6627578316 

Log.  67,  1.8260748027 

Lo.  3082         3.4888326343 


Log.  3083=3. 


6165 


NUMBERS  AND  THEIR  LOGARITHMS, 

OFTEN    USED    IN    COMPUTATIONS. 


Circumference  of  a  circle  to  dia.  1 }  Log. 

Surface  of  a  sphere  to  diameter  IV  =3.14159265  0.4971499 
Area  of  a  circle  to  radius  1  ) 

Area  of  a  circle  to  diameter  1  =  .7853982  —1.8950899 
Capacity  of  a  sphere  to  diameter  1  =  .6235988  — 1.7189986 
Capacity  of  a  sphere  to  radius  1  =4.1887902  0.6220886 


Arc  of  any  circle  equal  to  the  radius  =  57°29578  1.7581226 
Arc  equal  to  radius  expressed  in  sec.  =  206264"8  5.3144251 
Length  of  a  degree,  (radius  unity) =.01 745329  —2.2418773 

12  hours  expressed  in  seconds,      =    43200  4.6354837 

Complement  of  the  same,       =0.00002315  —5.3645163 

360  degrees  expressed  in  seconds,  =   1296000          6.1126050 

A  gallon  of  distilled  water,  when  the  temperature  is  62° 
Fahrenheit,  and  Barometer  30  inches,  is  277.  ^VV  cubic 
inches. 


,/277.274=  16.651 542  nearly. 


277'27 


=18.78925284  V  231  =15.198684. 


.775398 

_  J28~2~  =  16.  792855. 

282' 


._=  18.948708. 


.785398 

The  French  Metre— -3.2808992,  English  feet  linear  mea 
sure,  =39.3707904  inches,  the  length  of  a  pendulum  vi 
brating  seconds. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


MAR  2  0  1950 


APR     5  H 


50 


LD  21-100m-9,'48(B399sl6)476 


Vp 
'  L/ 


79^1007 


QR 

5SI 


